minimal ellipsoids and their duals

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Sewie 11, Tomo XXXVII (1988), pp. 35-64

MINIMAL ELLIPSOIDS AND THEIR DUALS

PETER M. GRUBER

It is proved that "most" convex bodies in IE d touch the boundaries of their minimal circumscribed and their maximal inscribed ellipsoids in precisely d(d + 3)/2 points. A version of the former result shows that for "most" compact sets in IE d the corresponding optimal designs, i.e. probability measures with a certain extremal property, are concentrated on d(d + 1)/2 points.

1. I n t roduc t i on .

Let K~ be the space of all non-empty compact subsets of the

d-dimensional euclidean space IE d. Denote by C the subspace of /C

consisting of all convex bodies in IE a. These are defined as the compact

convex subsets o f IE a with non-empty interior. Assume tha t /E , C and any

of their subspaces be endowed with their "natural" topology. It can be

induced by the metric 5 defined by

6(C,D)=max~sup i n f l x - y [ , sup i n f i x - y [ } forC, D E E , l x6C y6D x6D x6C

where [ denotes the euclidean norm on Is d. The metric 5 was first

defined and put to use by Hausdorff and Blaschke. A slightly generalized

form of Blaschke's selection theorem implies that E is a complete metric

space and C is locally compact ([26], p. 91). The same is true for their

36 PETER M. GRUBER

closed subspaces. A version of the Baire category theorem says that in a complete metric or locally compact space a" countable union of nowhere dense sets, i.e. a meager set, may be considered as being 'small'. We say that a property holds for most elements of such a space or is generic if it holds for all elements except those in a meager set ([13], w

In recent years many properties of convex bodies and compact sets have been found to be generic. For the case of convex bodies consult the survey [11] and also [35].

If C is an non-planar compact set in IE a, its minimal (circumscribed) or Loewner ellipsoid Ec(c) is the ellipsoid of minimal volume containing C. Its existence is trivial. The first published proofs of its uniqueness seem to be due to Behrend [2] (d = 2), Danzer, Laugwitz and Lenz [7] (d > 2) and Zaguskin [34] (d > 2). An interesting Helly type theorem which has applications in functional analysis .is t he following result of Behrend [2] (d = 2) and John [14] (d > 2): For each convex body C there are points p l , . . . ,pk in C, where k < d(d + 3)/2, such that Ec(C) = E~((pl , . . . ,Pk}) . For other results on Loewner ellipsoids see Firey [8]. In convexity theory the concept of minimal ellipsoid turns out to be a useful tool for the characterization of ellipsoids and euclidean spaces ([12]). It is of importance in the context of affine differential geometry ([23]), functional analysis ([10, 15, 24, 32]), optimal control ([5]), statistics ([22]) and, as recognized recently, for the optimization algorithm of Shor and Khachyan ([1, 17, 18, 21, 28]).

In section 2 we shall show that most convex bodies intersect the boundary of their minimal ellipsoid in exactly d(d + 3)/2 points. Several related results are mentioned without proof. As a byproduct we shall obtain new proofs for the uniqueness of the minimal ellipsoid and the theorem of Behrend and John mentioned before.

For a convex body C the maximal (inscribed) ellipsoid or inellipsoid Ei(C) is the ellipsoid of maximal volume contained in C. The existence of the inellipsoid is easy to establish, its uniquenesss was again proved by Behrend [2] (d = 2), Danzer, Laugwitz and Lenz [7] (d > 2) and Zaguskin [34] (d > 2). Behrend [2] showed that in the case d = 2 for each convex body C there are supporting halfspaces $1 , . . . , Sj, of C, where k < (d + 3)/2 such that Ei(C) = Ei(S1 f-) . . . A S k ) .

MINIMAL ELLIPSOIDS AND THEIR DUALS 37

Section 3 contains outlines of proofs of the uniqueness of the inellipsoid, the extension of Behrend's Helly type theorem to d _> 2 (theoreby confirming a remark of Danzer, Grtinbaum and Klee [6]) and, finally, a sketch of a proof of the following result: For most convex bodies C the inellipsoid intersects the boundary of C at precisely d(d + 3)/2 points. Several related results are mentioned. To a large extent the proofs of the results of section 3 are dual counterparts of the proofs of the corresponding results of section 2.

We next study so-called experimental designs in the context of the equivalence theorem of Kiefer and Wolfowitz, making use of an interpretation of Silvey ([16, 19, 20, 22, 29, 30, 33]). Let C be a non-planar compact set in IE d. A probability measure # on C for which the information matrix

M ( # ) = ( f c X X t r d # ( x ) )

is non-singular is called an experimental design. (Ix stands for trasposition and the points x i n IE d are considered as column vectors. Thus xx tr is a d x d-matrix and M ( # ) i s obtained by integrating each element of xx t~ over C.) The basic equivalence theorem of Kiefer and Wolfowitz says that for an experimental design u the following propositions are equivalent:

(i) det M(u) > det M(#) for any experimental design #,

(ii) max{xtrM(u)- lx : x 6 C} <_ max{xt~M(#)-lx : x 6 C} for any experimental design #,

(iii) max{xtrM(u)- lx : x 6 C} = d.

Let us call an experimental design u satisfying (i), (ii), or (iii) an optimal design. It was proved by Sibson that for an optimal design the

ellipsoid {z : zt M(u)-lz <_ d}

is the (unique) ellipsoid of minimal volume with center at the origin o and containing C. The support of u is contained in the intersection of C and the boundary of this ellipsoid.

An immediate consequence of a result of section 2 is that an optimal

38 PETER M. GRUBER

design of a generic compact set in IE d is supported by at most d(d+ 1)/2

points (see section 4).

The affinities which map a convex body C onto itself also map its minimal ellipsoid and thus also the set of their common boundary points

onto itself. Together with a result of section 2 this shows that for a generic convex body the group of affinities is finite.

In section 5 it is shown that for most convex bodies C the group of

affinities mapping C onto itself actually consists of the identity only.

The abbreviations diam, bd, int, conv and id stand for diameter, boundary, interior, convex hull and identity mapping, respectively, v()

denotes the volume, and Wd is the volume of the solid euclidean unit ball B d in IE d. The unit sphere in IE a is denoted by S d-1 . For x E IE a let

( x l , . . . , xd) *r denote its coordinate array, o() is the usual Landau symbol.

2. Minimal circumscribed ellipsoids.

We shall call a compact set in IE d irreducible if it is nonplanar and

if each proper subset either is planar, or else its minimal ellipsoid is of

smaller volume than the minimal ellipsoid of the given set.

THEOREM 1. Most convex bodies C in C have precisely d(d + 3)/2

points in common with the boundary of their minimal circumscribed ellipsoids. These points form an irreducible set with minimal circumscribed ellipsoid Ec(C).

Our proof of this result is rather lengthly and makes use of ideas

from projective geometry, the geometric theory of positive quadratic

forms and also of the implicit function theorem and of Carath6odory's

theorem on convex hulls. The basic idea consists of identifying ellipsoids in IE d with points in IE d(g+3)/2.

There are several supplementary results of the theorem of Behrend

and John and of Theorem 1. Of these we mention the following where the concept of irreducibility has to be adapted to the respective situations.

For each centrally symmetric convex body C there is an irreducible set of


at most d(d + 1)/2 pairs of points in C which are symmetric with respect

to the center of C and with minimal ellipsoid Ec(C). Most centrally

symmetric convex bodies C have precisely d(d + 1)/2 pairs of points in common with the boundary of EC(C) and these are symmetric with respect to the center of C. These points form an irreducible set with minimal

ellipsoid Ec(C). For a compact set C in IE a which is not contained in a

proper subspace of E d let E~(C) denote the (unique) ellipsoid of minimal volume which contains C and with center at the origin o of IE a. Call

E~(C) the o-minimal (circumscribed) ellipsoid of C. A compact set in

E a will be said to be o-irreducible if it is not contained in a proper subspace of IE a and if each of its proper subsets either is contained in a proper subspace or else has an o-minimal ellipsoid of smaller volume

than the o-minimal ellipsoid of the original set. For each convex body C there is an o-irreducible set of at most d(d+ i ) /2 points C, say p l , . . . , pk, k < d(d + 1)/2 such that E~(C) = E~({pl , . . . ,pk}) . Most convex bodies C have precisely d(d + 1)/2 points in common with the boundary of

E~(C) = E~({p l , . . . , p l } ) . Theorem 1 and the results mentioned in this paragraph hold for compact sets too. For later reference we mention only.

THEOREM 2. Most compact sets C in E are non-planar and have

precisely d(d + 1)/2 points in common with the boundary of their o-minimal

ellipsoid. The points form an o-irreducible set with E~(C) as its o-minimal

ellipsoid.

Since the proofs of the results in this paragraph are very similar to that of Theorem 1 and partly much simpler in detail we shall omit them.

Preliminaries. First, some geometric results are formulated, followed

by several results from the geometric theory of quadratic forms. Then

we shall represent ellipsoids in IE d as points of the euclidean space E. = IE d(d+3)/2. Using this representation we proceed to give further auxiliary

results needed for the proof of Theorem 1.

Let 0 denote the origin and �9 the ordinary inner product of E. Points

of E are denote by A, B , . . . , subsets by C, O, . . . .

The elementary proof of the following relative of the Farkas Lemma

40 PETER M. GRUBER

([31], p. 55) is left to the reader.

In E let there be given a convex cone

~ ' ~ { A : A . V < _ I } V E t/I

(1) with non-empty interior, and a hyperplane / '= {A : A �9 N = 1} containing the apex of the cone. Suppose that M is compact and that / I /and N are contained in a hyperplane of E not containing 0. Then the intersection of T with the interior of the cone is empty if and only if N E convM.

Next we prove the following proposition:

Let P be a non-planar set in IE a and assume that each ellipsoid which contains P and has o as an interior

(2) point has volume _> rod with equality holding for B a. Then there exists no ellipsoid of volume < tog containing P .

Suppose that this is not the case. Then there is an ellipsoid of volume

< wd that contains P and does not contain o in this interior. Hence there

is a halfspace containing P with o on its boundary. Without loss of

generality we may suppose that P C {x :l l < 1, z > 0). Note that for 0 < ~ < 1/2

:1 1 _< 1, x I _> o} C {x : ( x l - s )2+ (1 -2s ) ( (x2 )2+ -. -+(xd) 2) <__ (1--S)2}.

Thus the ellipsoid on the right hand side contains P , contains o in its

interior, and for all sufficiently small E > 0 has volume

(1 - z)a(1 - 2z)-(d-1)/2wd = (1 - s + o(s))we < we,

Since this contradicts the assumptions in (2), we obtain (2).

A quadric in IE d (a conic in case d = 2) is a surface which can be

considered as a level set of a quadratic polynomial on IE a.

t A quadric Q in IE a which contains d + 1 non-coplanar points (3) of S a-l, say p l , . . . , pa+ l , and which has the tangent hyperplanes

at these points in common with S a-~ coincides with S d-~.

Embed IE a into projective space IP d by adjoining the hyperplane at infinity.

The proof of (3) is by induction. For d = 2 (3) holds, see Brauner ([4]), p.


58 (4)). Let d < 3 be chosen and assume that (3) holds for d - 1 instead

of d. Let T be the hyperplane containing P l , . . . , P d and let E1,...,7"4 denote the common tangent hyperplanes of Q and S d-1 at p~ , . . . , Pal. Let

p be the intersection point of T1 , . . . , Td. Then T is the polar hyperplane

of p with respect to both Q and S a-1 and p ff T. By induction hypothesis

Q A T = S d-1 rq T . Thus the polarities in T which assign to each point of

T the intersection of T with its polar hyperplanes with respect to Q and S d-~ respectively, coincide. Note that for the common boundary point

Pd+l of Q and S d-1 we have Pd§ r T . From these remarks it follows that Q = S d-l , see Brauner ([4], p. 191).

The following remarks stem from the geometric theory of positive quadratic forms ([27]). An ellipsoid E in IE d with center at o can be

represented in the form

E = {x : x t rA 'x <_ 1}

where A' = (aq) is real positive symmetric d x d-matrix. We have

Wd (4) v(E) - (det A')I/2"

Instead of v(E) we shall also write v(A') or v(o, A'). The ellipsoid E or, what amounts to the same, the matrix A' can be represented by the point

( a 1 1 , . . . , aid, a 2 2 , . . . a 2 d , �9 �9 � 9 add) tr C IE g (d+ l ) /2 = E p.

Then, the set of all ellipsoids in IE a with center o is represented by

an open convex cone O' in E' with apex at the origin. Let I ' be the

d x d-unit matrix.

(5)

O' = {A ' C P' �9 v(A') <_ wd} is an unbounded closed strictly convex smooth subset of P' with nonempty interior. The tangent hyperplane T' of O' at I ' meets D' at I ' only and separates D' strictly from the origin of E p.

The boundary of D' is called the discriminant surface.

We shall now extend these remarks so as to cover also the case of

ellipsoids with midpoints ~r o. Let E' be embedded in the natural way

into E = IE d • E ' (last d(d + 1) coordinates).

42 PETER M. GRUBER

Any ellipsoid E in IE d with o C int E can be represented in the form

E = {x : 2atrx + x t rA 'x < 1}

where a = ( a 0 1 , . . . , aOd) tr E IE d and A' = (a,]) is a real positive symmetric

d x d-matrix. Since c = - A ' - l a is the center of E, we obtain

E = {x : (x - c)trA'(x - c) <_ 1 + at~A' - la} ,

thus implying that

(6) v(E) = (1 + atrA'-la)Ct/2Wd

(det A')I /2

We shall represent E by the point

( a o l , . . � 9 aod, a l l , . . . , ald, a 2 2 , . . . , add) tr C IF-- d(d+3)/2 = E.

Denote this point by A, (a, A'), or A(E); instead of v(E) we shall also

write v(A) or v(a, A'). The set of all ellipsoids in IE d which contain o in

their interior is represented by the open wedge P = IE d x P' . The origin 0

of E belongs to the set of apices of P. We say that a set is bounded away

f r o m 0 by its boundary if each ray starting at 0 either does not intersect

the set or else meets its boundary at a unique point and from this point

on is contained in its interior.

O = ( A E P: v(A) < wd) is closed in • and bounded away from 0 by bd D. We have bd D = {A E P : v(A) = wa) and bd D

(7) is a smooth surface. Let T denote the tangent hyperplane of bd D at (o, I'). Then Tfq D = {(0, I ')} and T separates strictly 0 and O.

The function v : A ~ v(A) on P is continuous, and by (6) it is strictly

decreasing on each ray in P starting at 0 if one moves away from 0.

Thus b d D = {A C P : v ( A ) = w d } and D is bounded away from 0 by

bd O. By considering the cases a = o, a 5/o separately one confirms that

v has nonvanishing gradient on P. Hence the implicit function theorem

yields that the level set bd O of v is a smooth surface. Since the inverse

of a positive matrix is also positiv, we deduce from (6) and (4) that

(a, A') C O implies A' C O' and, more precisely, that A' is an interior


point of O' in E' if a 5r o. Since by (5) we have T' M O ~ = {IP} it follows that (IE e x 7") N O = {(o, I')}. Hence the hyperplane IE e x T' coincides

with the tangent hyperplanes T of bd O at (o, I'). Noting that 7 "~ strictly separates O' from 0' by (7), we obtain that T strictly separates O from 0. This concludes the proof of (7).

As an application of (7) we shall show that

(8) f for each convex body C there exists a unique minimal ellipsoid.

The existence of a minimal ellipsoid of C follows from simple compacmess

arguments for the coefficients of the quadratic polynomials defining the ellipsoid containing G. If the minimal ellipsoid were not unique, there

would be two different minimal ellipsoids E , F , say. By applying a

suitable affine transform if necessary, we may suppose that E = B u. Let

(o, I '), (a, A') be the points in bd O corresponding to E = B u and F. For

all sufficiently small ), > 0 we have (1 -) , ) (o, I ~) + ),(a,A p) C i n t o by (7)

and thus

v((1 - ~)(o, I p) + )~(a, A')) < we = v(E) = v(F).

Since the ellipsoid corresponding to ( 1 - ) ~ ) ( o , I ~) + ) ,(a,A t) contains C

this contradicts the minimality of E , F , concluding the proof of (8).

A routine argument together with (8) shows that

(9) C, C1, C2, . . . E C, CI, C2 . . . ~ C implies Ec(Ci), E~(Cz),.. . ~ E~(C).

The proofs of (2) and (8) together yield that

(10) for any C E C the center of Ec(C) is contained in intC.

The mapping V : IE e ~ E defined by

V ( p ) = (2p l , �9 �9 �9 2p d, (pl)2, 2 p i p 2 , . . . , 2p lpd , (p2)2, 2 p 2 p 3 , . . . , (pd)2)tr E E

for p E IE e

is clearly related to the so called Voronoi-mapping in the geometric theory

of quadratic forms. It will play a crucial role in our proof of Theorem 1.

44 PETER M. GRUBER

An ellipsoid of the form {z : 2atrx + xtrA'x < 1} contains a point p E IE d

if and only if 2atrp + ptrA'p = A . V(p) < 1

where A = (a, A) E E and �9 denotes the ordinary inner product in E. Thus

the set of all ellipsoids in IE d containing o in their (11) interior and containing a given point p is represented in E

by the set {A : A . V(p) <_ 1} M P.

The following proposition will be used for the proof of the theorem

of Behrend and John and of Theorem 1.

Let C E C and suppose that Ec(c ) = B a. The set of all ellipsoids which contain C is represented by the set C M P

where_- f3 ca: A. v o)< 1}. (12) _ vebdC

6- is a closed convex set in E with non-empty interior.

K= r ) {A : A. V(p) <_ l} pEbdCf3bdB '~

is the supporting cone of C at (a, I ' ) and T separates K a n d D.

o E i n t C because of (10). Thus ( l l ) shows that the set of all ellipsoids

which contain C is represented by C M P. As an intersection of closed

halfspaces, C is a closed convex set in E. Consider an ellipsoid which contains C in its interior. Then all ellipsoids sufficiently close to it

still contain C. This shows that C has non-empty interior. Since C M P

represents the set of all ellipsoids containing C, and B d = Ec(c) , we have

v(A) > wg for all A E C ~ P , where equality holds if and only i f A = (o, I'). Hence CN D = {(o, I ')} and the tangent hyperplane r separates C and D by (7). For p E bdC f] bdB d we have (o, I ' ) . V(p) = (pl)2 + . . . + (pd)2 = 1.

ThUS (O, I') is a boundary point of the halfspace { A : A . V(p) <_ 1}

whenever p E bdC M dbB d. This shows that K is a convex cone with apex

(o, I'). Since all halfspaces appearing in the definition of K are closed, K

itself is closed too. Clearly K D C and the apex (o, I') of K belongs to

bd C. Supposing that K is not the supporting cone o f C at (o, I '), there is

an open ray starting at (o, I ~) contained in the interior of K which does

not meet C. Let R be a point on this ray. Then

(1 - (1/i))(o, I') + (1~OR ({ C for i E {1 ,2 , . . . } .


Thus, by the definit ion of O and since (o, 1') E O,

(o, I ' ) . V(qi) < 1 < (1 - (1/i))(o, I ' ) . V(qi) + (1 / i )R �9 V(qi) (13) for suitable qi E bdG and i E ( 1 , 2 , . . . } .

By choosing a subsequence and renumber ing if necessary, we may

suppose that ql, q 2 , . . . ---+ P E bdC and thus V ( q l ) , Y ( q 2 ) , . . . ---+ V(p) .

Hence (13) yields that (o, I~) �9 V(p) = (pl)2 + . . . + (pd)2 = 1 and therefore

p E bdC M bdB d. On the other hand, mul t iplying the inequalities in

(13) by i, gives (o, I ') �9 V(qi) <_ R �9 V(qi) and thus 1 < R �9 V(p). Since

p E bdC (q bdB d the definition of K yields R ~ intK. This contradict ion

shows that K is the support ing cone of O at (o, I~). Since the hyperplane

T separates O and O and contains (o, I ~) it also separates K and D,

concluding the p roof of (12).

We are now in a posi t ion to give a short p roof of the theorem of

Behrend and John.

Let C E C. Then there exists an irreducible set (14) { P l , . . . , Pk} E bdC fq bdEC(C), k <_ d(d + 3)/2,

such that Ec(C) = E C ( ( p l , . . . , P k } ) .

Without loss of generali ty we may suppose that Ec(C) = B d. Since 0 r T

by (7), we have T = (A : A �9 N = 1} for suitable N E E. The hyperplane

7" contains the apex (o, I ' ) of the cone K but does not intersect intK, see

(12). In particular N �9 (o, I ') = 1. The set M = {V(p) : p C bdC M bdB d} is

compact . V(p) �9 (o, I ' ) = 1 holds for p E bdC M bdB d. Hence both N and

t l / a re conta ined in the hyperplane ( A : A �9 (o, I ~) = 1} not containing 0.

An applicat ion of (1) then shows that

N C convM = conv{V(p) : p E bdC N b d B d } ( c {A : A . (o, I ~) = 1}).

Since the hyperplane {A : A �9 (o, I ~) = 1} has d imens ion (d(d + 3)/2) - 1,

a theorem on convex hulls ([3], p. 27) implies that

N is contained in conv ( V ( p l ) , . . . , V(pk)} for suitable pl, . . . , pk E bdC M bdB d, k < d(d + 3)/2.

N, V ( p l ) , . . . , V(pk) E {A : A �9 (o, I ~) = 1}. Hence we may apply (1) once

again to show that TO intL = 0 where L is the cone

k

L = N { A : A . V(pO ~ 1} i=l

46 PETER M. GRUBER

with apex (o, It). Since /. D K, int K 5/O, and since T separates K and D

the hyperplane T also separates/, and D. Noting that /'M O = ((o, F)} and (o, _r t) E /. we obtain/ , f3 O = {(o, F)}. Hence v(A) >_ Wd for all A E / - f3 P where equality holds if and only if A = (o, F). The ellipsoids which contain

P l , . . . , Pk and of which o is an interior point are represented by /. tq P,

see (11). Thus, an application of (2) shows that EC({pl,... ,pk)) = B d. If

{p l , . . . ,pk) is not an irreducible set one may delete suitable pi's such

as to obtain an irreducible set with minimal ellipsoid B d. This concludes

the proof of (14).

We say that two compact sets in IE d are "close" to each other if their distance in the sense of the metric 8 is "small".

Proof of Theorem 1. We first show that

the set of C E C for which bdC f3 bdEC(C) consists of at most (15) d(d + 3)/2 points is dense in C.

Let D E C. By (14) there is an irreducible set ( p l , . . . , pk) C bdDMbdEC(D) where k < d(d + 3)/2 and such that E~(D) = EC({pl,. . . , pk}). From

{Pl, . . . ,Pk} c c o n v { p l , . . . , p k ) c D(X) =

(1 - ,k)D + ,kconv{pl, . . . , Pk) C D

it follows that EC(D(s EC(D) for ~ E [0,1]. This together with

bdD(),) f3 bdE~(D)= ( p l , . . . ,pk} for ), E [0, 1] and the fact that D(),) is arbitrarily close to D if ), > 0 is sufficiently small, concludes the proof

of (15). Using(15) we shall prove the following refinement of (15).

(16) For most C E C the intersection bdC f) bdEC(C) contains at most d(d + 3)/2 points.

For i G (1 ,2 , . . . } let

Ci =(C E C" bdC n bdEC(C) contains at least (d(d + 3)/2) + 1 points

with mutual distances _~ 1/i}.

An elementary compactness argument employing (9) shows that Ci is

closed in C. By (15) Ci has empty interior and thus - being closed - is


nowhere dense in C. Hence

Ol3

U Ci is meager in C. i=1

Since

Oo

U Ci = {C E C" bdC Yl bdEC(C) contains at least (d(d + 3)/2) + 1 points}, i=1

(16) is proved.

The second part of the proof of Theorem 1 is more complicated, the most difficult part being the proof of the following proposition.

(17)

Let {pl , . . - , pk} be an irreducible set where k < d(d + 3)/2. Then there are irreducible sets arbitrarily close to {p l , . . . , pk}, having the same minimal ellipsoid as {pl , . . . ,pk} and consisting of exactly d(d + 3)/2 points.

Without loss of generality we may suppose that EC({pl, . . . ,pk}) = / 3 d Since {Pl , . . . , Pk} is irreducible, p l , . . . ,Pk E bdB d= S d-1. An application of (12) to the convex body cony {pl, ... ,pk} yields that

(18)

k

K = N { A " A . V(p]) <_ 1} j=l

is a closed convex cone with non-empty interior and apex (o, I'). The tangent hyperplane r = {A �9 A �9 N = 1} of O at (o, I ') separates K and O.

The next proposition is as follows:

(19) Let J~. {1 , . . . , k}. Then {pj - j E J} is contained in an ellipsoid which contains o in its interior and has volume < we.

If the points pj �9 j C J are coplanar, then (19) is easily seen to be true.

Suppose now that the points pj �9 j C P are not coplanar. If (19) is not

48 PETER M. GRUBER

true, (2) shows that Ec({Pj : j C J}) = B d, contradicting the irreducibility of {p l , . . . ,pk} . This proves (19). From (18)~ (7) and (19) we obtain that

(20)

f for any J ~ { 1 , . . . , k } the set N { A " A . V(pj) <_ 1} jcJ

is a closed convex cone with non-empty interior and apex (o, I ~) which contains K and is not separated from O by E In particular, 7" intersects the interior of this cone.

(21) N G conv { V ( p l ) , . . . , V(pk)}, but N ~ conv{V(pj) : j E J} for any J ~ { 1 , . . . , k } .

We next prove the following.

(22) Let U be a relatively open subset of S d-1. Then

V(U) C H = {A : A . (o, I ') = 1}, but V(U) is contained in no proper subplane of H.

U C S d-1 clearly implies that V ( U ) C H. Suppose now that V(U) is contained in a proper subplane of H. Since 0 ~ H, there is a subspace of E of the form {A : A �9 M = 0} with suitable M 5/0 which contains V(U). Thus V(x) �9 M = 0 for all x C U. Hence U is contained in the quadric {x : V(x) . M = 0}. Since U is a relative open subset of S d-1 this is possible if and only if this quadric coincides with S d-1 and thus, in particular, does not contain o, which is impossible. This proves (22).

(23) Let U 1 , . . . , Uk C S 'd-1 be open neighbourhoods of p l , . . . , pk relative to S d-1. Then N belongs to the interior of conv (V(U1)U. . . U V(UD) relative to H.

Note that V ( S d-l) C H and therefore V(UI) , . . . , V(Uk) C H. In order to prove (23) let us suppose that the conclusion of (23) does not hold. The there exists a halfplane in H containing V ( U D , . . . , V(Uk) and

having N as a boundary point relative to H. By (21) each of the points

(o,I ') C 7"= {A : A . N = 1} and p l , . . . , p k C bdB d imply that N , V ( p D , . . . , V ( p k ) c H{A : A . (o,I ') = 1}. Thus (1), (18) and (20) show that


V'(pl) C V ' ( U 1 ) , . . . , V ( p k ) E V'(Uk) must belong to the boundary relative to/-/of this halfplane. Since 0 ~ H, this halfplane can be represented in the

form {A : A . L _< 0} A H with suitable L ~ 0. Hence V ( p l ) , . . . , V(pk) are

boundary points of the halfspace {A : A �9 L _< 0) and V(U1) , . . . , V(Uk) are contained in it. This means that the quadric Q = {z : V(z) �9 L = 0}

contains p l , . . . , pk while U~, . . . , Uk(C S d-l) are located on the same side of it. Thus the tangent hyperplanes of Q and ,_qd-1 at the common points

P l , . . . , Pk coincide. Since { p l , . . . , pk} is an irreducible set, p l , . . . , pk are non-coplanar. Hence Q = S d-1 by (3) and thus o g Q, contrary to the

definition of Q. This proves (23).

Let e > 0. Since {P l , . . . , Pk ) is an irreducible set with EC({pl , . . . , pk)) = B d, we may choose open neighbourhoods U1, . . . , Uk C S d-1 o f

p~ , . . . , Pk, respectively, relative to ,_qd-1, such that diam Uj < e /2 and

(24) v(EC(U1 U . . . U Uj-1U Uj+I U " " U Uk)) '( rod for ] C {1 , . . . , k}.

A consequence of (23) is that

there are points ql, . . . , qt G U1 U �9 .. U Uk such that N is an (25) interior point of conv {V(ql ) , . . . , V(qt)}(C Iq) relative to H.

This will be refined as follows:

(26) f There are points S l , . . . , s ,~ C U1 U - . . U Ok, ra = d(d+ 3)/2

such that conv ( V ( s D , . . . , V(sm)}(C/4) is a simplex and N is an interior point of it relative to H.

Call a simplex of dimension < (d(d + 3')/2) - 2 degenerate if it contains N. Let t be the number of degenerate simplexes with vertices in

{V(q l ) , . . . , V(ql)}. If t = 0 the proof is finished by (25). Suppose now

that t > 0 and denote by conv {V(qj , ) , . . . , V(qj,)} a degenerate simplex

with vertices in { V ( q 0 , . . . , V(qt)} and having minimal dimension. Then N

is an interior point of this simplex relative to its affine hull. Since V(Uj,) is

not contained in the affine hull of this degenerate simplex by (22) we may

choose a point q C Uj~ close to qj, such that the following are fulfilled: N

is an interior point of conv{V(ql ) , . . . , V(qj~-l), V(q), V(qj~+l),. . . , V(ql))

relative to H furthermore N ~/conv{V(q), V(qj2! . . . , V(qj,)}; and, finally,

a simplex with vertices in {V(ql ) , . . . , V(qj,-1), V(q), V(qj,+l) , . . . , V(qt)}

5 0 PETER M. GRUBER

can be degenerate only if the corresponding simplex with vertices

in {V(ql) , . . . , V(qt)} is degenerate. Here "corresponding" means having the same vertices except, possibly, for V(q) which is to be replaced by V(qjl). Thus we have obtained the following results: N is an interior point of conv{V(qD, . . . , V(q), . . . , V(qt)} relative to H, further ql, ... ,q, . . . ,ql E U1U .-. U Uk and the number of degenerate simplexes with vertices in {V(ql),..., V(q),..., V(qt)} is at most t - 1. Repeating this step a finite number of times one arrives at the following. There are points r l , . . . , r t E U~ U. . . U Uk such that N is an interior point of conv {V(r l ) , . . . , V(rt)} relative to H and there is no degenerate simplex with vertices in {V(rD, . . . , V(rt)}. This implies (26).

(26) shows that N E conv {V(sD, . . . , V(s,~)} and N ~ conv {V(Sy) : j E J} for any J ~ { 1 , . . . , m}. Hence (1) implies the following:

(27)

Consider the cone

K = N ' [ A : A . V(si) < 1} j= l

with apex (o, I'). Then (o, I ') E T and r n intK = O.

(28)

t Let J ~ { 1 , . . . , r e } . The cone

Kj = ["]{A : A . V(sj) < 1} D K jeJ

~ has apex (o, I ') and rMintKs :! 0.

Note that ?" strictly separates 0 and D and TM/:1 = (o, F) by (7). (27) shows that 0 E K, further that 7"M int K = 0 and that (o, F) C ?'M K. Thus 7" separates K and D and KM D = (o, I'). From this, together with (11), it follows that all ellipsoids with o as an interior point and containing Sl , . . . , s~ have volume > Wd, where equality holds for B a only. Now an application of (2) shows that EC({Sl, . ,s,~})= B a. Let J ~ { 1 , . . . ,m}. Then (28) together with (7) implies that Kj Mint D ~ 0. Hence (12) shows that there is an ellipsoid of volume < rod containing {sj : j C J}. These facts show that {Sl , . . . , s,~} is irreducible with minimal ellipsoid B a. Note

that S l , . . . , s,~ E UI U. �9 �9 UUk. If some Uj did not contain any of s l , . . . , sm

MINIMAL ELLIPSOIDS AND THEIR DUALS 5 1

then v(EC{sl, . . . , am})) < cod by (24), which gives a contradiction. Hence

for each Uj there exists an si C Uj. Since the Uj's are neighbourhoods of

the pj ' s relative to S a-1 and since diana Uj < e /2 , the distance (in the

sense of the metric 6) between { p l , . . . , p k } and {s l , . . . , sm} is less than

e. Since e > 0 was choosen arbitrarily we thus have finished the proof of

(17).

Having shown (17), the remaining part of the proof is routine. The

first step is to show that

(29) the set of C C C for wich bdC f'l bdEC(C) is an irreducible set consisting of precisely m = d(d + 3)/2 points and with minimal ellipsoid E~(C), is dense in C.

Let D C C. By (14) there exists an irreducible set { p l , . . . , p k } C bdD Cl bdEC(D) with E~({pl, . . . ,pk}) = EC(D). By (17) there is an irreducible

set {Sl , . . . , sm} close to {Pl, . . . ,Pk} for which EC({Sl , . . . ,Sm})= E~({Pl , . - - ,Pk}) = EC(D). For small ), > 0 the convex body

D()0 = (1 - ),) conv (D U {81, . . �9 , 8m}) + ~k conv {81, . . . , 8 m }

is close to D and has EC(D) as its minimal ellipsoid. Since bdD(),)C) bdE~(D) = {Sl , . . . ,sin} this proves (29). Using (29) we shall show that

(30) for most C E C the intersection bdC tq bdE~(C) contains an irreducible set consting of m = d(d + 3)/2 points and with minimal ellipsoid Ec(C).

For each

and with

Let

C E C there is an irreducible set consisting of at most m points

minimal ellipsoids Ec(C) contained in bdC f'l bdE~(C) by (14).

Co = {C C C : bdCt~ bdEC(C) contains an irreducible set of less

than m points and with minimal ellipsoid Ec(C)}.

In order to show (30) we shall prove that Co is nowhere dense in C. The

first step is to prove that C0 is closed. For this let

( 3 1 ) C1,C2,...ECo C I , C 2 , �9 . . ---~ C E C.

52 PETER M. GRUBER

We have to show that C C Co. By the definition of Co there exist

irreducible sets

(32)

{Pll,...,Plkl} C bdC1 f'l bdEC(C1) where kl < m and EC( {p11, . . . , Plkl }) = Ec(C1),

{ P 2 1 , . . . , P 2 k 2 } C bdC2 71 bdE~(C2) where k2 < m and S C ( { P l l , . . . , Plkl }) = Zc(C2),

(9) implies that

(33) EC({p11, . . . , P l k l } ) = Ec(c1), EC({p21, . . . ,P2k2}) = Ec(C2), ...--> Ec(C) �9

By considering a suitable subsequence and re-indexing if necessary we

may suppose that k 1 -- k2 -- . . . . k, say, with k ~ 7T~ and

(34) P11,P21, . . . - * Pl, . . . ,Plk,P2k, . . . --* Pk

where p l , . . . ,pk C bdCf' l bdE~(C) by (31), (32), (33). If pl, . . . ,pk were

co-planar, then (33) would not hold, giving a contradiction. Thus p l , . . . , Pk

are not co-planar and therefore

E C ( { P l l , . . . , Plk} ) , EC({P21, . . . , P 2 k } ) , . . . ---* E C ( { p l , . . . , Pk})

by (34) and (9). From this together with (33) we deduce that

EC({pl , . . . ,Pk}) = Ec(C) . The set {Pl,...,Pk} or suitable subset of its is an irreducible set in bdC Cl bdEC(C) of at most k < m points and

with minimal ellipsoid Ec(C) . This proves that Co is closed in C. Hence

intO0 = 0 by (29). Thus Co is nowhere dense in C, concluding the proof

of (30).

Theorem 1 now readily follows from (16) and (30).

3. Maximal inscribed ellipsoids.

A set of closed halfspaces in IE d will be called irreducible if the

following conditions hold: (i) Their intersection is a convex body. (ii)


The intersection of the halfspaces of a proper subset either is unbounded or else contains a maximal inscribed ellipsoid of larger volume than the intersection of the original set of halfspaces.

THEOREM 3. For each convex body C in C there is an irreducible

set {$1 , . . . , Sk} of k < d(d + 3)/2 supporting halfspaces o f C such that

E~(C) = Ei(S1 N . . . N Sk).

THEOREM 4. Most convex bodies C in C have precisely d(d + 3)/2

supporting halfspaces in common with their maximal inscribed ellipsoid.

These halfspaces form an irreducible set with maximal inscribed ellipsoid

E~(C).

As a consequence of Theorem 4 we obtain that most convex bodies

have precisely d(d + 3)/2 boundary points in common with their maximal inscribed ellipsoid. Below we shall outline proofs of the uniqueness of the

inellipsoid and of Theorem 3 and 4 which are "dual" to the corresponding

proofs of section 2. Additional tools from projective geometry will be

needed.

As is the case for Theorem 1, there are also several related results

of Theorem 3 and 4. The following facts, perhaps, deserve mentioning.

The concept of irreducibility has to be adapted in a suitable manner. For each centrally symmetric convex body C there is an irreducible set of at

most d(d + 1)/2 pairs of supporting halfspaces, symmetric with respect

to the center of C such that their intersection has inellipsoid Ei(C).

Most centrally symmetric convex bodies C have precisely d(d+ 1)/2 pairs of supporting halfspaces symmetric with respect to the center of C in common with their inellipsoid. These halfspaces form an irreducible set

and the inellipsoid of their intersection is E~(C). For a closed convex set C with o E in tC which contains no line there is a unique ellipsoid E~(C) C C with center at o and having maximal volume. For each convex

body C with o 6 int C there is an irreducible set of supporting halfspaces,

say $ 1 , . . . , Sk, k <_ d(d + 1)/2 such that EL(C) = E~(S1 Q . . . Q Sk). Most convex bodies C with o 6 in tC have precisely l = d(d + 1)/2 supporting

halfspaces in common with E6(C), say $1 , . . . , St. These halfspaces form

an irreducible set and E~(C) = Ei(S1 fq.. �9 fq St). No proofs of these results

54 PETER M. GRUBER

will be given.

Outline of the proofs of Theorem 3 and 4. We shall state and partly

prove a series of propositions that will lead to Theorem 3 and 4 and which, in some sense, are "polar duals" of the corresponding results in

section 2. This is also indicate in the numbering by adding a *

(1") = (1) with A replaced by /3 .

Let S be a set of closed halfspaces in Ig d, the intersection of which has non-empty interior. Assume that each ellipsoid

(2*) which is contained in this intersection and which contains o in its interior has volume < wa where equality is attained for B d. Then there is no ellipsoid of volume > Wd contained in this intersection except for B d.

(This is slightly more general than the corresponding result (2).) Our proof

of of (2*) is modelled along the lines of Danzer, Laugwitz and Lenz [7].

Let E be an ellipsoid of volume > Wd and contained in the intersection of

the halfspaces of S. By choosing an appropriate orthonormal coordinate system we may represent B d and E in the form

[zl[ <~__ COS 01 ...COS Od_2COS 0d-1

]Z21 __~ COS 61...COS 6d-2 Sin 6d-1

Ixa-ll _< cos 61 sin 62

[ where 01, . . . C 0,

I d] sin 01

( IxlI--<~OQCOS01'''COS~d-2COS6d-I+fll [ 7r] 1 x : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , where 01, . . . C 0,

I~dl ~_~ Ot d sin 01 +/~d

~vhere oq, . . . ,oea > 0, oq.. .oee _> 1, and fl l , . . - , /3d > 0, are suitably chosen. For ~ E (0, 1) define the ellipsoid E0 , ) to be

Ixll < (1 - ,k + ,ko~l)cos Ol . . . COS 6d-2 COS Od_ 1 + ~/~1 "~ . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . , w ere ) Izel _< (1 - ,k + ,koee) sin 01 + 3d

MINIMAL ELLIPSOIDS AND THEIR DUALS 5 5

In the fol lowing the products are to be extended f rom 1 to d. We have

_> ( , - x +x w,,

where equality holds in the first inequality if and only if 1 - ), + ),oq =

. . . = 1 - - X + ~Ot d a n d t h u s a l = . . . = Otd ( s e e e.g. [25], p. 192) and in the second inequali ty i f and only if ~ a j = 1. (Note that ~ a j _> 1 by

assumption). Hence v(EO,)) > wd unless E is a translate of B a. Since

o C in tE()Q and - by its definition - E 0 , ) is contained in the intersection

of the halfspaces of S for small ), > 0, we conclude that E is a translate

of B a. If E ~ B a, then it is easy to specify an ell ipsoid in conv (B d U E)

of vo lume > wd and with o in its interior. Since this ell ipsoid is also

contained in the intersection of the halfspaces f rom S, a contradict ion to

the assumptions of (2*) results. Hence E = B a, concluding the p roof of

(2*).

(3*) = (3),

(7*) = (7) with A replaced by B.

An immedia te consequence of (2*) is that

(8*) each convex body C has a unique maximal ellipsoid.

From this one can deduce

(9*) = (9) with E c replaced by E i.

Let E be an ell ipsoid with o E int E and represent bdE in the form

bdE= {x : 2atrx + xtrA'x = 1}. p

Then the support ing hyperplanes o f bdE are precisely the hyperplanes of the form {x : utrx = 1} for which 2btru + ut~B% = 1 and where the

matrices (--1 atr~ -1 (bl btr~

a A ' J ' B ' ] '

coincide up to a mult ipl icat ive constant. I f b = (b01, . . . , bod)tL B' = (bi])

we clearly may represent E by the point

(b01, . . . , boa, b l l , . . . , bld, b 2 2 , . . . , b 2 d , . . . , bd~) tT E E.

5 6 PETER M. GRUBER

E is contained in a halfspace S(u) of the form {x : utrx <_ 1} (u 5r o) if and only if

2btru + ut~B'u = B �9 g(u) < 1.

Thus

(10") the set of all ellipsoids in IE e with o contained in their interior and contained in a given halfspace S(u) is represented in E by the set {B : B . V(u) < 1}.

Having stated (10") we continue our considerations. The set

{u : 2btru + ut~B'u < 1}

is the familiar polar reciprocal of E with respect t o Sd-1; we denote it by E*. For later reference we shall prove the following:

(11") Let E be an ellipsoid in E d with o C int E. Then v(E)v(E*) >_ w E.

(Note that this is not contradiction with a well-known inequality of Blaschke and Santal6 which say that for an o-symmetric convex body C we have v(C)v(C*) < to E, where equality holds precisely for ellipsoids.) By appropriate choice of a coordinate system we may suppose that E has center (,./1,..., ,./d) and principal axes parallel to the. coordinate axes and of lengths 251,. . . ,25 a. Then {x : z1 = ~1 -I-- 51} are supporting

hyperplanes of E. Hence the ellipsoid E* contains the line segment with endpoints (1/(,-/1- 51), 0 , . . . ,0), (1/(,,/1 +51), 0 , . . . ,0). Thus E* intersects the xl-axis in a line segment of lenght 251/((51) 2 - (,,/1)2) and similarly for the other axes. Steiner symmetrization of E* in all coordinate planes gives an ellipsoid with principal axes on the coordinate axes having lengths at least 251/((51)2 __ ( ,~1)2) , . . . , 25d/((Sd)2 _ (5d)2) . Thus

51 v(E)v(E*) _> 5 1 . . . 5dwd(51)2 - (,.),1)2

5 d w 2 ,

(5e) 2 _ (,~d)2 t~ _>

concluding the proof of (11").


Using the representation of ellipsoids described in the foregoing paragraph, one deduces from (7)*,(10)*,(11)* the following.

Let C G C and suppose that Ei(C) = B d. The set of all ellipsoids contained in C and with o in their interior is represented by the set G* n P where

C*= N {B:B.V(u)<_ I} S(.) supporting halfspace of C

(12") 6"* is a closed convex set in E with non-empty interior.

K*= N {B:B.V(u)<_I} 8(,0 supporting

halfspace of c and B d

is the supporting cone of C" at (~ I9 and r separates K* and D.

(12"), (1"), Carath6odory's theorem, (10") and (2*) combined yield

Theorem 3.

The next proposition is a simple consequence of Theorem 3 and (9*).

For most C E C the bodies C and Ei(C) have at most (16") d(d + 3)/2 common supporting halfspaces.

The halfspaces considered below are always assumed to contain o in their interior and are represented in the form S ( u ) = { z : u t"z < 1}. The most difficult step in the proof of Theorem 4 is to establish the following proposition:

Let ( S ( u D , . . . , S(uk)} be a minimal set of closed halfspaces where k < d(d + 3)/2. Then there are irreducible sets {S(wl) , . . . , S(w,,,))} consisting of exactly

(17") ra = d(d + 3)/2 halfspaces with {Wl, . . . , win} arbitrarily close to {Ul , . . . , uk} and such that E ~ ( S ( u l ) n . . . n S ( u k ) ) = E ~ ( S ( w l ) n . . . n S(w, , , ) ) .

The proof of (17") is achieved along the following lines. We may suppose that E i ( S ( u l ) M . . . M S(uk)) = B a. Then similar proofs as in sect. 2 lead to

r N E conv{V(u l ) , . . . ,V(uD} , (21") l X ~ c o n v { V ( u j ) ' d E d } for d ~ { 1 , . . . , k } .

5 8 PETER M. GRUBER

(22*) = (22) with A replaced by B.

(23*) = (23) with p l , . . . , pk replaced by u l . . . , u k .

Let z > 0 and choose open neighborhoods Uj of uj in S a-1 with diam U] < e/2 for j C {1 , . . . , k} and

(24*) ( forjV(Ei(N{S(u)'c {1 , . . . , k}.u C U1 I,.J... I,,J Uj -1 [,3 Uj+I [ ,J . . . I,.3 Uk})) > t, Od

(26*) = (26) with s l , . . . , s,~ replaced by w l , . . . , w,,.

From this one can deduce (17") using (1"), (7"), (11"), (12"). Rout ine continuity arguments show that

for most C E C the bodies C and E~(C) have an irreducible (30*) set consisting of m = d(d + 3)/2 supporting halfspaces in

common and the intersection of the halfspaces has maximal ellipsoid E'(C).

Theorem 4 now follows form (16") and (30*).

4. Optimal designs.

As remarked in the introduction, the support of an optimal design of a non-planar compact set C in IE a is contained in bdC N bdES(C). Since the non-planarity of a generic compact set C is easy to prove, Theorem 2 yields

THEOREM 5. Most compact sets in IE a are non-planar and the

supports of their optimal designs consist of at most d(d + 1)/2 points.

5. Groups of affinities of convex bodies.

For a convex body C let .A(C) denote the group of affinities which map C onto itself, i.e. the group of symmetries of C. We shall show that most convex bodies are "asymmetric", more precisely we prove


THEOREM 6. The convex bodies C in IE d with A (C) = (id) form a

dense open set in C.

We first put together some needed results which are either well

known or easy to prove. Let A , / 2 and (.9 denote, respectively, the groups

of affine, linear and orthogonal transformations of IE a onto itself.

(35) Let C E C and assume that Ec(C) has center o.

. Then bdC 0 bdEC(C) contains a basis of IE d.

(36) Let C E C. Then A(C) C A(Ec(C)).

For (36) see e.g. Danzer, Laugwitz and Lenz [7]. A sequence of affine transformations of IE d is said to converge if for some fixed basis of

IE d the matrices corresponding to the affine transformations converge

elementwise.

j ' L e t C, C 1 , C 2 , . . . E C and A, A 1 , A 2 , . . . E.4 be such that (37) 1

l C1 ,C2 , . . . ~ C, A I , A 2 , . . . ~ A. Then A I C 1 , A 2 A 2 , . . . ~ AC.

For L E /2 let (/ij) be the corresponding matrix with respect to some

orthonormal basis of IE d. Then

is independent of the particular choice of the orthonormal basis. For the

6 0 PETER M. GRUBER

next proposition see e.g. [9], p. 124�9

For each L E CO there exists an orthonormal basis of IE e such that to L corresponds the matrix

(IX. \

�9 "1. "--1

(38) - 1

"COS ~01 -- sin ~91

sin ~DI COS ~DI

~os~k -s in~ok sin ~ok cos ~ok

where blocks of one or two types may b e missing and the elements not indicated are all 0.

In order to prove Theorem 6 we show at first that

(39) 79 = {C E C : .A(C) 5t {id)} is closed in C.

Let C1, C2, . . . C 79 and suppose that

(40) C1, C 2 , . . . ---r C E C.

We have to show that C C 79. Without loss of generality ~ve may

suppose that B e ( C ) = B e. From (40) and (9) we obtain that EC(C1),

Ec(GD,... - , l?,c(C). Hence there are A1,A2,. . . C A with

(41) A I , A 2 , . . . ~ id

and E~(C.i) = A j B a for j C {j = 1 , 2 , . . . } . F r o m Cj C C and (36) it follows that

{ id}~Af 'A(Cy)Aj C AflA(Ec(Cj))Ay = O.

Let Lj C AflA(Cy)Ay \ {id}. With respect to a suitable orthonormal basis

the orthogonal transformation Lj 7~ id is represented by a matrix as shown

in (38). A suitable power of L j, say

(42) Mj C Af lA(Cj)Aj C (.9,


then satisfies I I M j - idll > 1. We may choose a subsequence, say

M j , , M j 2 , . . . , which converges to an orthogonal transformation M.

Clearly IIM - idll _> 1 and thus M 7L id. (40), (41) and the fact that Mjl , M j 2 , . . . 4 M together with (37) show that

A j l M j l A ~ I C 1 , Aj2Mj2Af21C2,.. .---* M C .

On account of (42) this may b e written in the form C1, C2, . . . --~ M C . Thus (40) yields C = M C or M E A(C) . Since M 7fid we obtain that C C 7:), concluding the proof of (39).

The second step of our proof consists of showing that

(43) C \ D = {C C C : A ( C ) = {id}} is dense in C.

Let P be a polytope. Without loss of generally we may suppose that

E c ( P ) = B d. Using (35) choose a basis pl, . . . ,Pd C bdP ('1 bdB e. The points P l , . . . , P d are vertices of P . Let E > 0 be so small that the following hold:

1 (44) O < e < - ~ m i n { l p - q l : p , q v e r f i c e s o f P , psCq} (<1 ) .

(45) L e t A O be such t h a t l A p i - p i l < s f o r i c { 1 , . . . , d } . Then I IA- idll < 1.

Choose q l , . . . , qd C bdB d such that

(46) E > IPl - qll = 21192 - q21 = " " = 2d- l lPd-- qdl > 0

and consider the polytope P(e, qi) = conv (P t2 {q l , . . . , qd}). Then P(z, qi) differs from P by most s in the sense of the metric 6. Let

A E A ( P ( s , qi)) C .A(B d) = O. The orthogonal transformation A maps the

set of vertices of P(E, qi) onto itself and does not change distances. Hence (44), (46) yields

A ( { p l , ql}) = {Pl, q l } , . . . , A({pd, qg}) = {pg, qd}.

From this, and (46) we deduce that A satisfies the assumption of (45)

and thus IIA - idll < 1. Hence we have proved that I]A - idll < 1 for all

62 PETER M. GRUBER

A C .A(P(e, q0). Since .A(P(e, q0) is a group, an argument which makes use of (38) implies that .4(P(e, ql)) = {id} and therefore P(e, qi) ~ 79, thus proving (43).

(39) and (43) together yield Theorem 6.

6. Acknowledgement.

I should like to thank Professor Zamfirescu for a discussion on Baire type theorems in convexity, in which he suggested the problem of determining the precise number of the common boundary points of a generic convex body and its Loewner ellipsoid. I am indebted to Professor Brauner and Dr. Sorger for pointing out various results from projective geometry. Professor Brauner, in particular, provided the short proof of proposition (3). Finally I am obliged to Professors Groemer and Schnitzer for their valuable remarks.

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Pervenuto il 4 settembre 1986, in forma modificata il 12 gennaio 1987.

Institut ffir Analysis, Technische Mathematik und Versicherungsmathematik

Technische Universit?it Wien A-1040 Vienna, Wiedner Hauptstr. 8-10

minimal ellipsoids and their duals

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