approximation of convex bodies by polytopes

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie I], Tomo XXXI (1982), pp. 195-225

A P P R O X I M A T I O N O F C O N V E X B O D I E S B Y P O L Y T O P E S

PETER M. GRUBER - - PETAR KENDEROV

Let C be a convex body of E a and consider the symmetric difference metric. The distance of C to its best approximating polytope having at most n vertices is 0 (1/n2/(a-1)) as n ~ oo. It is shown that this estimate cannot be improved for any C of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of <~ most ~ convex bodies are rather irregular and that for d = 2 << most ~ convex bodies have unique best approximating polygons with respect to both metrics.

1. In t roduc t ion .

A convex body is a compact convex subset of d -d imens iona l Eucl idean

space E d having non-empty interior . Let ~ denote the space of all convex

bodies of E d and let l1 ]l be the Euc l idean no rm on E a. The HausdorH-metric

on ~ is defined as fol lows:

(C, D ) : = max { max min I lx - yl[, m a x min I lx - yl]} for C, D E ~ . xEC y~D y6D xEC

If Ix denotes Lebesgue measure on E a and A the symmetr ic d i f ference then the

symmetric di//erence metric 0 on ~ is defined by

0 ( C , D ) : = I x ( C A D ) for C, D E ~ .

Both metr ics induce the same topology on C. W e assume that ~ is e ndow e d

wi th this topology. For n E bi let ~n denote the subspace of ~ consis t ing of

all polytopes of ~ having at most n vert ices. (g', is empty for n < d).

196 PETER M. GRUBER - PETAR KENDEROV

The approximation of convex bodies by polytopes is important as a tool for various other investigations (see for example [2], [3], [7], [12], [21]), but also for its intrinsic interest (see [8], [10], [29] and the references there). For each CE E and nEN there are polytopes P.=P. (C), O .=O. (C)~E ~. . called best approximating polytopes, such that ~ (C, P.) = ~ (C, ~.) and 0 (C, On) = = 0 (C, ~.). Here

(C, ~.) : = inf { ~ (C, P)] P E ~n },

0 (C, $,) : = inf { 0 (C, Q) I 0 E ~,, }.

Typical approximation problems deal with the following:

(i) Upper and lower bounds for 8 (C, ~'n), 0 (C, ~',) for C'E ~ or for C belonging to special classes of convex bodies.

(ii) Asymptotic behaviour of 8 (C, ~,), 0 (C, ~',) as n--, oo

(iii) (Asymptotically) extremal convex bodies.

(iv) Approximation of ~ typical ~> convex bodies and uniqueness problems.

Analogous questions arise if one considers polytopes with a given number of facets or inscribed and circumscribed polytopes.

In many approximation problems it is no essential restriction if instead of ~ (C, ~,) one investigates inf { ~ (C, P) [ P E ~n, P c C }. We will use this without explicit reference. For d = 2, 3 many results are known. See for example Eggleston [8], Fejes T6th [10], McClure and Vitale [19] and Popov [22]. Dudley [7], Bronstein and Ivanov [4] and Betke and Wills [1] independently

proved that for each C'E

const (1) ~ (C. $.) < ~ for all n E N,

n2/(d-1)

where the constant depends only on the circumradius of C. On the other hand a result of Schneider and Wieacker [26] shows that

const (2) ~ (C, ~'~) > ~ (> 0) for all n,E N

n2/(d-1)

if C E ~ is of class ~2 having positive principal curvatures. A further result of Schneider [25] yields

const ( C , ~ ) ~ ~ a s n - - - oo

n 2 / ( d - l )

APPROXIMATION OF CONVEX BODIES BY POLYTOPES 197

for C E ~ of class ~3 and with positive principal curvatures. The constant is specified in terms of the Gaussian curvature of C. For d = 2, 3 this has been indicated by Fejes T6th [10], p. 43, 119, 149.

By the Blaschke selection theorem (see for example [2], p. 62 or [12],

p. 154, 201) ~ is locally compact. Hence the Baire category theorem implies that sets of first category in ~ are ~ small ~ compared with their complements (see [13], p. 132, 133).

A result of Schneider and Wieacker [26], proved independently by us (see below), shows in essence that for each monotone decreasing function

1: Iq---, R + with limit zero

(C, ~,) < ! (n) for infinitely many n

for each C E ~ except those convex bodies belonging to an F~-set of first

category in ~.

For 0 only a few general results are known, but for d = 2, 3 works of Eggleston [8], Fejes T6th [10], McClure and Vitale [19] and Popov [22] contain a wealth of results. Perhaps one reason for this is that a polytope O which is close to a convex body C with respect to 0 is not necessarily close to C in the neighbourhood of a particular boundary point of C. Thus, in contrast to the situation in the case of ~, much less information about O is available. Also, in contrast with ~, the general approximation problem for 0 and the approximation problems for inscribed and circumscribed polytopes are not closely related to each other. A result of Macbeath [18] shows that for each n amongst all convex bodies of equal volume the ellipsoids are worst approximable by inscribed polytopes of $ , . Dudley [7] shows that an immediate consequence of (1) is the following: For each C E~ we have

const (3) 0 (C, g'n) --< ~ for all n E Iq,

t /2 / (a -1)

where the constant can be chosen such that it depends only on the circumradius of C. A simple alternative proof can be given using the above mentioned

result of Macbeath [18] and estimates for balls.

This paper contains the following results: For CE ~ of class ~2

0 (C, g',) >__ const

n 2 / ( d - 1) ( > O) for n Elq,

where the constant depends only on bounds for the curvatures of C (section 2).

1 9 8 PETER M. GRUBER - PETAR KENDEROV

If f, g : Iq ~ R + are such that 0 < / < g = o (1/n 2~(a-l)) as n ~ o o , then for

all convex bodies, except those from an F,-set of first category in ~,

(C, g',), 0 (C, g'n) < / (n) for infinitely many n,

(C, g'n), 0 (C, ~ ) > g (n) for infinitely many n

(section 3). For d = 2 all convex bodies, except those from an F~-set of first category in ~, have unique best approximating polygons Pn (C), On (C) for each n (section 4).

Most of our results can be transferred to convex functions defined on convex sets or on polytopes of E d. There are also corresponding results for circumscribed or inscribed polytopes.

The following notation will be used throughout the paper: For A, B c E a and LEI t define A + B : = { x + y l x E A , yEB} and ~,A: = { ~ , x l x E A } . Let diam, bd, int, conv, pos, vert and schL stand for diameter, boundary, interior, convex hull, positive hull, set of vertices and Schwarz symmetrization with respect to a line L. For the latter see [2], p. 86, or [30]. ~ denotes ( d - 1)- dimensional surface area and ~ Lebesgue measure in E d. The symbols for Euclidean norm and inner product are ti"11 and ( - , - ) and o, U denote the origin and the unit ball of E d. o (.), O (.) are the Bachmann-Landau symbols.

2. Very smooth convex bodies cannot be approximated too closely by polytopes.

THEOREM 1. Let C E ~ be a convex body. Then there is a positive constant

depending only on d and the circumradius of C such that

const (1) 0 (C, ~n) <- ~ ]or all n 'E N.

n2/ (d -1)

Suppose /urther that C is of class ~2, having positive principal curvatures. Then there exists a positive constant depending only on d and the maximum and the minimum o] the principal curvatures o] C such that

const (2) 0 (C, ~n) >- ~ /or all n E N.

n 2 / ( d - 1)

(1) is known (see the introduction). We mentioned it only for reasons of com- pleteness. Analogous results hold if one considers polytopes with at most n facets or inscribed or circumscribed polytopes. Essentially the same proof shows that for the validity of (2) it is sufficient that the boundary of C contains an


arbitrarily small piece of a ~2-surface of positive principal curvatures. In parti-

cular one may cancel the assumption of Theorem 1 that the principal curva-

tures of C are positive.

PRELIMINARIES. First we put together some tools which will be used in

the proof below.

A (Euclidean) ball of radius 9 is said to roll in a convex body C if for

each point of bd C there is a ball of radius 9 contained in C and containing

that point. A ball of radius ~ can roll around C if for each point p E b d C there is a ball of radius r containing C with p on its boundary. Blaschke [2],

p. 118, and Koutroufiotis [17] proved the following:

(3) Let C E ~ N ~ have positive principal curvatures (curvature ]or d = 2).

Then the following propositions are equivalent:

(i) The principal curvatures o] C have lower bound 1/r and upper

bound 1/9.

(ii) A ball (disk ]or d = 2 ) ot radius 9 can roll in C and a ball o]

radius r can roll around C.

For C'E ~ N C ~ and p E bdC denote by n (p) the exterior normal unit vector

of C at p. A normal section of C is the intersection of C with a 2-dimensional plane containing some p EbdC and also p + n(p). If C is of class ~2, then

also each of its normal sections is of class ~2.

(4) Let C E ~ N ~2 have positive principal curvatures and suppose that

1/r and 1/9~ are lower and upper bounds respectively o] the principal

curvatures o] C. Then 1/r and (9] + 4~)1/2/9~ are lower and upper

bounds o] the curvature ol each normal section.

By formulae of Euler and Meusnier (see [9], p. 154, 152) it is sufficient for the proof of (4) to show the following:

(5) Let C N N be a normal section o] C and let q be a point o] the relative

boundary of C N N. Then the cosine o] the angle ~ between n(q)

and N is at least pff(p] + 4 ~ ) ~.

Choose p E b d C such that p, p + n ( p ) E N . By (3)

(6) p -- 91n (p) + pl U c C c p -- r (p) + r U.


Let n (q) = n + m where n is parallel to N and m orthogonal. If m = 0, then

n (q) is parallel to N and (5) holds trivially. Now suppose m # 0. For each

x E C the inequality 0 _< ( q -- x, n (q)) holds. Since (6) implies

x: = p - - p l n ( p ) + ~ pt

llmll m E C,

we thus have

P l (7) 0 __< (q - - p + p tn (p) - - m, n + m) .

Ilmll

Because m is orthogonal to N, it follows f rom p -- p~ n (p), q E N that ( q - - p +

+ p ~ n ( p ) , m ) = 0 . The right inclusion of (6) and p - - p t n ( p ) , qE C imply

I l q - P + p l n ( p ) ] l - < 2 r Thus (7) yields

p, l imtl<-(q-p-l-p~n(p), n )<- l lq -p+p~n(p) l l llnll<2~,llnll.

This shows that tanq~ = ItmIl/llnl; <_ 2r Hence c o s 9 ->- p,/(pz + 4 ~ ) ~ ,

concluding the proof of (5). This finishes the proof of (4).

(8) Let ~i : [0, [3g] ~ R, (i = 1, 2), (0 < [31 --< [32) be two convex ]unctions o] class C 2 with 9i ( 0 ) = q~/ ( 0 )= O. Suppose that ]or the curvatures

k~ of ~ the inequality 0 < k2 < kl holds on [0, [31] and that from a point (0, - v) (v > O) there exist tangents to the graphs of tpi, meeting

the graphs at points with abscissae "el. Then "~ < "~2.

Since v = -- c.~ (-r~) + ~i' ('~i) "~, we obtain

v = f ( - - 9i ( '0 + 9i ' ( '0 "r)" d ' r = f qoi" ('r) "~ d % o 0

and thus

"r 'T2

(9) f qh" ('r) ": d ' r = f 9~" 0:) "r d ' r . 0 0

The formula

(10 ) k i - - (1 + r

will be used below. Let d/i: [0,[3i] ~ [0,~/2[ be chosen such that t a n f f i = % ' .

APPROXIMATION OF CONVEX BODIES BY POLYTOPES 2 0 1

Then (10) implies

sin ~bi ('c) = f ki ("Q d -c. 0

Since kl _> k2 > 0 on [0, ~31], we conclude that sin kbl >-- sin ~b2 >__ 0 and thus qh' ---- tan +l --> tan ~/: = q~2' on [0,~31]. Hence (10) and k l _ > k : > 0 yield qh" >--q~" on [0,[3a]. From (10) and k2 > 0 on [0,~2], it follows that q~2" > 0 on [0, [32]. This combined with (9) yields "~ _< "~:, concluding the proof of (8).

Let A c E d be compact and p EE ~. Each point l E A for which

[[1-- Pl] = in f{ l lx - - PI[ [x EA}

is called a metric projection or a loot of p on A. If A is compact and convex, then for each p E E d there exists a unique foot I on A. In this case the mapping p ~ f is called the metric projection of E d onto A.

(11) Let C E ~ be such that a ball of radius > 9 can roll in C and denote

by D the set o] all centres o] balls o] radius 9 contained in C. Then

the Iollowing hold:

(i) D is compact and convex.

(ii) Each point pE C \ i n t D has a unique loot on bdC and a unique

loot on D, say ] and g. For ], g we have II] -- gll = p and pE conv {1, g).

It is obvious that (i) holds. To prove (ii), let p E C \ i n t D be chosen. D is the set of all centres of balls of radius 9 contained in C. In C a ball of radius > p can roll. Thus p has a unique foot on bdC, say ], and the following holds:

(a2) Ill - PlI -< P, p = t - l i t - P l ] n ( t ) .

The point ]E bd C has a unique foot on D, say g. Again using the fact that

D is the set of all centres of balls of radius 9 contained in C, we have

I l l - - gtt = 9, ( t + 9 U ) n D = , [g } , g = ] - - Itg - - I t] n (]).

From this and (12) it follows that II/-gll = p, p E c o n v { / , g } , and g E ( p +

+ l l g - p i i U ) N D c ( ] + [] ] - -gl iU) N D = { g } . Thus g is t h e unique foot of p on D. This proves (11). A lemma of Busemann and Feller shows that

(see [20], p. 35)

(13) for each CE ~ the metric projection o] E d onto C does not increase distances o] points.

2 0 2 P E T E R M . G R U B E R - P E T A R KENDEROV

A result of Dinghas [6] says that the Steiner symmetrization of measu- rable sets of E d does not increase their distance with respect to the metric 0. In particular this holds for convex bodies. Since the Schwarz symmetrization of a convex body can be considered as the limit of a sequence of Steiner symmetrizations in particular hyperplanes (see Blaschke [2], p. 86, Wills [30]) and the measure tx is continuous on C, the result of Dinghas implies that

(14) /or all C, D E ~ and each line L o/ E d the inequality O(C,D)>__ _> 0 (schL (D)) holds.

Our last preliminary result is the following:

(15) Let l E E d, where II]11 = 9, /urther ~E]O, 1] and C, D E e be chosen. Suppose

{x I Ilxl[ -< 0, tan (x, 1) <_ 9} c c c {xl (x,]) <_ pz}

and o as well as (1 + 92) � 8 9 Then O(C,D)>__ ~1~ ~+~, where ~ depends only on d and 9.

Here (x, ]) denotes the angle beween x and ].

e l

Figure 1

Let L be the line through o, f. It follows from (14) that

0 (C, D) > 0 (schL (C), schL (D)).

Together with C, D also schL(C), schL(D) fulfill the assumptions of (15).


Obviously 0 (schL (C), schL (D)) is greater or equal to the measure of the body obtained by rotating the shaded area in the figure about L. Let tOk denote the k-dimensional measure of the k-dimensional unit ball. Choose the parameter "~ according to the figure. Crude estimates show that

0 (SChL (C), schz (D)) __

--> (d -- 1)rod-1

= p d ( d - 1) tOa-l([3 (1

fl((l+~2) V2 --1) \a-2 p2.~2 ~ (1+~2) ~ ) 2 ~ (1 +132) ~ +

tOd-i (p ( ( 1 + ~32) ~A - - 1 - - ' 0 ) a

+ - - - d - - - ~3 ~-I =

~ 2 )d-2 ,T2 +[32)�89 ((1+ ~32) �89 +1) 2 ~ (1 +~2) ~ +

+ 9a tOd-1 ((1+~ 2) -- (l+x)2) a > d ~a-t((1+132)~ +1 +,~)a --

> 9a ( (d-- 1) rod-1 .r2 ~d-3 t,.Od-1 - - 2 �9 4 e -1 + d �9 3 d

(I +~2--(1+'r ) > ~ d -1

>

pa (d-- 1 ) (.0d-1 1~ ~ d - 3 - - 2 - 4 a-1 1 + -- 1 for ( l + ' r ) 2> 1 + - -

~ 2

4 >

1 for (1 + '~)2< 1 + ~ ~a-1 4

I 9a ( d - l ) toa_l 9atOd_l lf3a+l. > min 2 �9 4 a+3 ' d 74 ~

This proves (15).

PROOF OF THEOREM 1. We have to prove the second half of Theorem 1 only. Choose C E ~ N ~2 with positive principal curvatures. Let 1/0"1 , 1 / 9 i be the minimum and the maximum of the principal curvatures of C. Define

o : = ( p ] + 4 ~ ) ~ ' ~ : = 2 ~ 1 .


In the following ~ , ~ . . . . . n~, n2 . . . . denote positive constants depending only

on 9, a, d. By (1),

(16)

We show that

(17) after a translation

n 6 N, n >_ nl .

~2 0 ( C , O , ) < - - for h E N .

n2/(d-l)

we may assume that p U c C, O, c ~r U !or all

By (3) a ball of radius P I ( ~ 2p) can roll in C and a ball of radius o'2(= o'/2)

can roll around C. Hence after a translation

2pU c C c (~r-- 2 p ) U

holds. Suppose pUqt Q.. Then 2pUcC yields

0 (c, O.) _> ~ (2 p U \ Q ~ ) > f~3.

This contradicts (16) for n >_ n2 : = (~32/~33) (a-1)/2. Hence

p U c O , for n 6 N , n>__n2.

Suppose now that O, q to 'U. Then C c ( a - - 2 p ) U and p U c Q . for n___n2

imply

0 (C, O,) >_ tx ( O , \ C ) >_ Ix ( 0 , \ ( 0 - 2) U) > [34 for n 6 N, n _> n2.

This is in contradiction to (16) for n > n3: = ([3z/~1) (d-')/2. Hence

Q, c C U for n 6 N , n > n 3 .

This proves (17) for n , : = max{n2, n3}.

(18) Let v 6 v e r t Q , \ C and denote by [ its /oot on C. Then [ I f - - v i i - <

< p2/(100o-)/or all n 6 N, n >__ n4.

The inclusions p U c Q, c o" U, v 6 Q, and C c { x ] < x, n (])) < (/, n (f)) } imply

0 (c, o . ) >__ ~ ( o r \ c ) _ ~5 II1 - vii.


Hence (16) yields

~2 I l l - vii < - - ~5 n 2/(d-l) ~

thus proving (18).

(19) Let v E vert O n \ C and denote by f its toot on C. Then f -- p n (f)E O.

for all n E N, n >_ns.

For suppose ] - - p n (f)(~ O. . Since a ball of radius p can roll in C, we have

0 (c, On) > ~ ( ( / - pn (f) + p U) \On) >--~ (p U)/2.

This contradicts (16) for n >__ ns : = (2 ~z/~ (p U)) (d-1)n. Hence ! -- p n (f)E O. for n > ns.

We consider first the case

(20) n >__ max{n1, n4, ns}.

The definitions of p, o- and (3), (4) imply that

(21) a bail of radius p can roll in C and a ball of radius o- can roll around C.

Each normal section of C has curvatures between l/o- and 1/p.

A simple consequence of (17) is the following:

(22) Let " denote the radial projection of E d \ { o } onto bdC. Then ~(A ' ) > --

_> (p/o')o~ (A) for each Borel set A c bd Qn.

The first important step in the proof is to show that a good proportion

of bdC is contained in Qn:

(23) Let B: = Qn n bdC. Then tz(B) >_ 96tz(bdC).

The sets AI: = ( b d Q n ) \ i n t C , A2: = (bdQn) t3 C are Borel sets of b d Q , and A1 U A2 = bd Q, . The strict convexity of C implies that

(24) ~ (A1) + (~ (A2) = t/, (b d Qn).

It follows from 0 (C, Q.) = 0 (C, ~.) that

0 (C, Qn) _ 0 (C, (1 + e) O,) for all ~'E R +.

206 P E T E R M . G R U B E R - P E T A R K E N D E R O V

Since by (17) and (21)

0(C, (1 + e) O.) ___ 0 (C, O.) + [o ' t t (AI) -- Eptz (A2) + o(E)

we thus obtain

An application of (24) gives

o r

(At) >-- p-- o~ (A2). o"

as ~--- + 0,

0~ (A3 + p-- ~ (A1) --> p (a (A2) + tz (A3) = p - - - - ~ ( b d O n ) r f f (2"

P tz (A1) >- ~ ~ (bd On).

p+o"

Since B = AI' and bd C = (bd On)', it follows from (22) that

p2 p ~z (B) >_ ~ o~ (At) >-. ~ (bd On) >--- o~ (bd C).

o" o- (p +o-) o-2 (p +o-)

This proves (23).

Let vl . . . . . vk denote the vertices of Q, not contained in C. For each

v~ let ]i be the foot of v~ on C. Denote by Bi the set of all boundary points

p of C which can be seen from vl, that is, C f ) c o n v { v i , p} = {p}. Since

every point of B can be seen from at least one of the vertices v~, the inclusion

B c Bt U . . . U Bk holds. Thus (23) shows

(25) (B1 U . . . U B~) >_ 96~ (bdC).

For 16 bd C let t signify the orthogonal projection of E d onto the tac plane

of C containing ]. Define the (spherical) cap C (1, ~) of centre ] and radius [3 by

We prove,

(26)

c G I 3 ) : ={pEbdCl/ can see p and l l t -p , ll

for each ] 6 b d C and each Borel set A c C(] ,p /3 �89 the following

inequality holds: o~ (A) <_ 2 e~ (Af).


Let /EbdC. It is sufficient to shows that ( n ( / ) , n ( p ) ) > _ l / 2 for each

pE C ([, 9/3�89 Choose p. The tac plane of C through p is either a tac plane

of the ball ] -- 9 n (/) + 9 U c C or does not meet it at all. Hence the parallel plane through pe does not intersect the ball and an elementary consideration

shows that (n (1), n (p))_> 1/2. This proves (26). The same argument proves

that, for each ]'EbdC and each rectifiable curve contained in C (f, 9/3~), the lenght of the curve is at most two times the length of its projection onto

the tac plane through f. This shows in particular the following proposition

holds where 3" denotes the geodesic distance on bdC:

(27) For each /E bd C and p E ~ (1, 9/3 �89 the inequality T (1, p) <__ 2 []/--p'][

holds.

Obviously,

(28) ]or 1, p E bd C the inequality T (1, p) <_ 9 implies that p E C (1, T (f, p))

and

(29) /or J, p E bd C the inequality []l-pl] ~ p implies that p'E C (1, 3" (l, P)).

The next proposition gives an idea about the size of the B, in terms of ][t,-v, ll:

(.30) C(h (p [],h-vd]/2) ~) c Bi c C ( h , (3or [Ih-v,]l /or iE{1 . . . . . k}.

Let N be a 2-dimensional plane containing 1~, vi. The two tac lines of C N N containing v~ define a convex arc on C fq N which consists precisely of those

points of C contained in N and visible from vi. Denote the tac points by

p • and define similarly tac points q• (]i -- p n ([~) + p U) N N and r • (/i +

--crn(/ i) + a U ) M N. The convex curves

( b d ( l i - - p n ( / 3 + p U ) ) N N, (bdC) N N, ( b d ( / , - - r N N

contain ]i, have a common tangent at ]i, and are of class C 2. If k denotes

the curvature of (bd C) N N then by (21) the inequality 1/~r ___ k <__ 1/9 holds.

Hence an application of (8) yields

(31) [IJi - q• ~ ]l.h - p• ~ [[I, - r+"[].

I l l , - q-'-',]l = p(llS'-v'll= § 2p I I / , -v, l l )~ p + I I / , -v , II

208 P E T E R M . G R U B E R - P E T A R KENDEROV

Since I I / ' - v, II-< p-<~ by (18), a simple computation shows that

p (2 p II/,-,,,11)~ 2p

= (-p l l / '2v ' l l ) ~

I1 t , - r-'-',ll = ~ + 2 ~ IIt,-~,11) ~ + II / ,-v,I I

< ,~ (3 o I I / , -v, ll) ~ _ I

O"

= (30-II / ,-v, ll)~.

Together with (31) and the definition of caps this proves (30).

Without restriction of generality one may assume that

(32) l i t , - ~111 ~ IIs~- ~11 ~ .. . ~ I I /~ - ~,11.

Choose indices l = i ~ < i 2 < . . . < i z _ _ - k as follows: Let i~: = 1 . Define iz to be the first index i > il such that

t, ~ c (t,,, (~ l i t , , - v,,l l/2)~).

Define i3 to be the first index i > i2 such that

t, ~- c( l , : , (p I l l , , - ~,,11/2) ~) u c(t,~, (p II/,~- u,~ll/2)~)

and so on.

The next step in the proof is to show that

k !

(33) u c ( / , ; (3o-11/,- ,~,11) '~) ~ u oct,,, 5 (o-I1~,,,- v,,ll)~). i = I / = 1

The definition of i~ . . . . . it implies that

(34) s,, . . . . . s,~-, ~ c (t,,, (p I l l , , - ,,,,1112)~),

s,~ . . . . . S,~-,EC(I,,, (p I l l , , - ,,,,1112)~) U c(l,~, (p I I & - ",~11/2)~), �9 �9 * ~ . . . . . . . . . . * �9 �9 �9 * . . . . ~ �9 �9 �9 �9 �9 , + �9 �9 �9 ~ ~ �9 . o

/ , , , . . . , t ,~ c (#,,, (p l i t , , - ,~,,11/2)~) u ~,. u c (/,,, (p I I / , , - ,',, 11/2)~) �9


To prove (33), let p E C (/,, (3 o" [I/~ -- v, l 1)~4) be choosen. If i = i] for some ], we are finished. Now suppose that for suitable ] we have i i < i < ii+t or ij < i in case j = I. Then (34) yields

(35) fi'E C (~ira, (p lll'm -- v,mlll2) ~) for some m <_ ].

Hence the definition of caps shows

I I / , ~ - ,if" -< (p I[/,m - v ,~l] /2)~.

Since by (18) the right hand side of this inequality is less than p/3 ~, proposition (27) yields

(36) T (r _< 2 Ill, m-/~ ' - I I -< 2 (p II.~,,,, - v , , , , l ] /2 )~ .

In the same way as (35) implies (36), it follows from

pE C ( f i , (S o" ] l . / / - ,',1 I) ~) and im ___ ij < i that

~, (/,; p) _< 2 (3~ I[f, m - v,mll)~.

This together with (36) and p _< o" gives

T (fl,m, p) _< 5 (o" Ilhm - v,~ll ~.

By (18) the right hand side is less than p. Hence it follows from (28) that

concluding the proof of (33).

We now construct a system of disjoint subsets of C and show at first the following:

(37) The caps c,,: = c (/,,~ (~ II/ , , - v,,11/32)~), J ~ { 1,...; l} are pairwise disjoint.

For suppose the contrary. Then there is some point p E C, i fq C~ m for suitable

j < m. By (17) the inequalities

(~ II/,,- v,,11/32) ~, (P II/,m- v,mll/32) ~ -< p/3~


hold. Hence p E Ci i fq Cim and (27) imply

"r qii; v) <- (p ]]li, - vi i [[/8)~, "7 (/ira, P) <-- (P l[lim - V im I I/8)~.

Thus by (32)

"r (lii, lira) <-- (O ]I/ii-- vi~]l/2)~.

Since by (18) the right hand side is certainly less than P, proposition (28) yields

/im~ c (ti,, (p I l t , , - v , , l l /2)~) ,

contradicting the choice of the indices. This proves (37).

(38) The sets Dii: = c o n v ( { l l , - - p n ( ] i , ) } U Cfili) U {(6t/,+ pos(n(/i/)))},

I E { 1 . . . . . l} are pairwise disjoint.

Because of (37) it is sufficient to prove the following:

(39) Let p ~ . D i i . Then each loot of p on bd C belongs to Ci i .

Consider two cases:

(i) p E C fl Di i . In C a ball of radius 2 p can roll and the distance from p to the boundary of C is at most p. We thus can apply (11). Choose ], g, D as in (11). The points g and f i i - - pn(]ii) are the feet on D for each point of the segments S: = c o n v { f , g } and T: =conv{] i i, fl i - p n ( ] i i)} respectively. Obviously

(40) (g, n (tii) ) <_ O, (1, n (10 } < (tti, n ql i) ).

If g = f i i - p n ( ] i i ) then ]=11 i by (11) and thus ]ECt i. Thus (39) holds.

Suppose now g ~ ]ii - - o n (]ii). The point p belongs to the segment S and

the cone K: = c o n v ( { l i i - - p n ( ] i i ) } U C~ij), the axis of which is T. If / E K /

then ]ECI i and (39) is proved. Suppose now ] r K. It follows from ] r K,

g ~ ]ii - - p n (]ii) and (40) that there are points s E S \ { g } and t E T of minimal

distance among all pairs of points from S and T. Hence

Ilg - ( t l , - p n (ti,))ll > Ils - tll.


On the other hand g and ]i s- p n (]ij) are the fcct of s, t on D. Hence (13)

implies

J i g - ([i,- Pn(/ii))[[ <- - l l s - tll.

This contradiction shows that ] q~ K cannot hold.

(ii) p • C l q D i i. Write p = q + L n ( / i i) where q ECzfllJ and - - p <

< L < + o o . Let r : = / i i for ~ , < 0 and r: = / i i + L n ( / i i) for ~,>__0. It is

elementary to show that

lit - VII (P II/ i , - vi,11132) .

The foot of r on C is /ij. Let I denote the foot of p on C. Then

r[/i;- /11-< (p IlXl,- vi,[l/32)

by (13). Since by (18) the right hand side is less than p, it follows from (29) that ]ECi i . Hence (39) holds in the second case too, finishing the proof

of (39) and thus of (38).

We prove,

(41) O(C 0 Di i, O,, fl D#) >_ ~7 ]]]ii - - ]Iij[](d+l)/2 for ]E {1 . . . . . I}.

We shall apply (15) with /ii; /i i - o n ( l @ / i i+ ((1 + J32) ~ -- 1 ) p n ( t @

CNDi j , O, f lDi i in place of / , o, (1+[32) ~ t, C, D where [3: =([[[ii --vii[[/32)w.

The assumptions of (15) arc fulfilled: By (18) wc have ~E]0 ,1] . From

]ii -- p n (]i i ) .Af_ p U c C, f i i '~ bd C N bd (/ii -- 9 n (/i i) + p U)

one obtains

(fi i -- pn( / i i) + 9U) fl D i i c C I"1 Di i c {x I ( x , n ( / i i ) ) ~ ( f i i, n (hi)) }.

(19) and the definition of [3 yields

[ii -- O n ([i i ) ~ Qn ~ Di i ,

tij + ((1 + ~2)~ _ 1) p n (li i) c cony {tii -- p n (ti,), vi i } c 0,+

Hence an application of (15) proves (41).


Since our aim is to find large lower bounds for 0 (C, Q.), we show that

l

(42) ~" Ill,,- vi, ll "-',~ >- 9~. j = l

From this together with (41) the desired bound will be obtained. The ine-

quality 5(uIIl~j-v~jll)~_< p/3 ~ follows from (18). Hence (25), (30), (33)

and (26) together imply

1 * 1 k ~x (bd C) "< 9"-"~ 17, ( U ni) "< i=1 - - -~6 6(. (i=lU C (]i, (3 O" II/, - v, li)~)) <

1 l

/=I

2 i

(5 (= 11/,, ",,11)~) ' - ' ~~ = -< 96 j=~-3 2 �9 5 a-~ COd-~ 0 "(d-l)/2 l

96 ,=~ ][/ii -- vir

thus proving (42).

It follows from (38), (41), the convexity of the function 'T" ->~ (d+l)/(d-1)

on [0, + oo [ and (42) that

O(C, On) >__ O(C n Dil, On n Dil) + . . , + O(C N Di,; On n Diz) >_~

!

, ~: (11/ , ; - ~,,11"-"'~) `~+'''~-" > 9, ~: I i t , , - v,,ll"+l'/" = 9,1 ' = '

i=x 1 >_

!

>_

9,1 9~ "+'/`~-') 9, >_ 9, ltd+l)/(a-l) - - ~z / ( , - l ) n21Cd-l) - -


This holds for n >__ max {nl, n4, ns} (see (20)). Since 0 (C, O.) is a decreasing function of n we conclude that

O (c, Q.) >__

thus proving Theorem 1.

n2/(d-1) for n'E N,

3. The approximation properties of typical convex bodies are irregular.

THEOREM 2. Let f, g : N ---, R + satisfy 0 < ! (n) < g (n) = o (1 /n 2/~-~)) as

n---, oo. Then for all convex bodies C E ~, except those from an F,-set of first category in ~,

(1) (C, 8.) < ! (n) for infinitely many n,

(2) (C, 8 , ) > g (n) for infinitely many n

and likewise for 0 (C, 8,).

Analogous results hold for inscribed and circumscribed polytopes and also if one replaces 8, by the set of polytopes having n facets. Moreover it is possible to replace ], g by countably minorized and countably majorized classes of functions without any additional complication.

From the proof of the Baire category theorem one may see that a convex body C satisfying (1) and (2) and the analogous inequalities for 0 can be <<constructed, as follows: We make use of the fact that for polytopes P obviously ~ (P, 8 , ) = 0 (P, 8 , ) = 0 for all sufficiently large n, and that for C E ~ N ~2 by 1 (2) and 2 (2) the inequalities ~(C,~.), 0(C,~,)___ >_ const/n 2/(d-l) ~ 0 hold for all n. Let pi be a convex polytope. By smoothing the edges slightly one can obtain a convex body C 1 of class ~2. In C l one inscribes a convex polytope p2 which approximates C 1 very closely. By smoothing pz slightly we can obtain a convex body C 2 of class ~2 and so forth. If in this process p1, C a, />2, C 2 . . . . differ only very little, then C : = P~ n c ~ n n p z n c 2 N . . . is a convex body satisfying (1) and (2). Alternatively one can obtain (for d = 2) convex bodies which fulfill (1) and (2) by integrating particular Cantor-like functions.

PROOF OF THEOREM 2. The proof makes use of 1 (1), (2) in the case of 8 and of 2 (1), (2) in the case of 0. Since the proofs for ~ and 0 are verbatim the same we consider the case of g only.


The function C ~ 8 (C, 8.) for C'~ ~ is continuous and ~. is closed for each n'E N. Thus the sets

{ c E ~ 18 (c, ~.) __< g (n) },

are closed for each n'E N.

Hence the sets

(5) ek : = { c e c 18 (c, ~.) ~ g (n) for n = k, k + 1 . . . . } =

= ~ { c ~ e [ 8 (c, 8.) <_ g (n) }, . = k

9 k : = { C E ~ i S ( C , 8,) >_ t (n) /or n = k, k + 1 . . . . } = oo

-- n { c E e 18 (c, 8.) >_ 1 (n) } n=Ir

are closed/or each k,E N.

We show that

(4) @k, fgk are nowhere dense in ~ /or each k'E N.

Because of (3) it is sufficient to prove that '~k, Ok have empty interiors. Suppose first that i n t ~ ~ 6 . Then there exists a convex body C E ~k N ~2 with positive principal curvatures. Hence 8 (C, 8,) >_ const/n 2/~d-1) > 0 by 1 (2). We have g ( n ) = o ( 1 / n 2/~d-1)) as n ~ oo by assumption. Thus C ~ ~k. Con-

tradiction. Suppose now that int Ok ~ q,. Then there exists a convex polytope PEfgk . Then 8 (P, $ , ) = 0 for all sufficiently large n. This contradicts the

definition of ~)k. Hence (4) is established. Since

~k = {C,E ~ [ 8 (C, 8 . ) < g (n) for all but finitely many n}, k = l

oo

U ~k = { C E ~ [ 8 (C, 8,0 >--] (n) for all but finitely many n}, k = l

we obtain

e \ ( ~ ~ku u Dk)= k m l k = l

= { C E ~ [ 8 (C, ~.) > g (n) for infinitely many n and

8 (C, 8 , ) < ] (n) for infinitely many n}.

Together with (3) an (4) this proves theorem 2 for the case of 8.


4. Typical convex bodies have unique best approximating polygons.

THEOREM 3. Let d = 2. Then for all convex bodies C E ~, except those /rom an F~-set of first category in ~, and ]or each n E N the best approximating convex polygons o/ ~n with respect to the metrics ~ and 0 are unique.

We strongly conjecture that the clause d = 2 is superfluous. For general d it is possible to prove analogous theorems for inscribed and circumscribed polytopes with n vertices or n facets. The proofs are even simpler than the ones given below.

A trivial example of a planar convex body having unique best approximating polygons of ~ for each n with respect to the metrics 3 and 0 is provided by a triangle. A further example is the semi-circle, at least in the case of the metric 3. This can be proved by means of proposition 2 of Kenderov [16] which says that for the metric 3 each best approximating n-gon of a planar convex body has a sort of ~ equioscillation property>~. (This is in analogy to a property of best approximating polynomials of degree n of a real continuous function on an interval of FI with respect to the Chebyshev norm). The semi circle seems to provide also an example for the case of the metric 0.

Figure 2

In the proof of the theorem the metrics 3 and 0 will be considered sepa- rately. We have tried to organize both cases so that the existing analogies become apparent.

Convergence in the sense of the metrics 3 and 0 respectively will be

denoted by ~ and

PROOF OF THEOREM IN THE HAUSDORFF METRIO CASE. In this section approximation will be understood in the sen~: of the metric 3. For k, n E N define

~ , : = { C E ~ [ C has best approximating polygons P, QE g, such that

3(P,Q) ~ 1/k}.


(1)

Our first aim is to thow that @k. in closed in ~. It is easy to see that

/or C E ~ and n >_ 3 a point or a line segment approximates C worse

than some polygon o] 8 , .

The continuity of 8, the Blaschke selection theorem, the fact that the limit of a sequence of polygons of 8. either belongs to 8. or is a line segment or a point and (1) combined yield the following proposition:

(2) Let C,C~,C 2 . . . . 'E~ be chosen such that C1, C ~ . . . . & C and let p~,p2 . . . . 'ES, be best approximating polygons o/ Ct, C 2 . . . . . Then

there exists a subsequence P~,, Pi2 . . . . converging to a best approxi-

mating polygon P'E 8. o] C.

This together with the continuity of 8 implies that

(3) ~k, is closed /or all k, n,E N.

The second part of the proof consists in showing that @k. is nowhere dense in @.

(4) Let C, DE ~ and PE 8~ a best approximating polygon ol C be given such that 8 (C, D) + 8 (D, P) = 8 (C, P). Then each best approximating

polygon O E ~. o ] D is also a best approximating polygon o/ C.

We have 8 (D, P) = 8 (D, ~,,).

For if not, then 8 (D. 8 . ) < 8 (D, P) and thus

8 (C, 8,,) < 8 (C, D) -I- 8 (D, 8,) < 8 (C, D) + 8 (D, P) = 8 (C, P) = 8 (C, 8,0.

Contradiction. Hence

{QES. ] 8(D,O) < 8 (D, 8,)} c

c {OES. 18 (C,D) + 8(D,Q) <__ 8(C,D) + 8 (D,P)} e--

c { Q E S . I S ( C , Q ) ~ 8 ( C , P ) } = {QES~IS(C,Q)--< 8(C, 8.)}.

Since the first (last) set is the set of best approximating polygons QE ~, of D (C) this proves (4).

APPROXIMATION OF CONVEX BODIES BY POLYTOPES

Note that

~ ( C , D ) = m i n { k > O I C c D + kU, D c C + k U } for C, D E ~ .

We prove,

(5)

217

let C E C and • E ~ a polygon such that some vertex w of O is con-

tained in bd (C + ~ (C, O)U). It Q'E C is obtained from O by cutting

off a subset of • sufficiently near to w, then C c O ' + 8(C,O)U still holds.

If there

w is the trivially.

(6) O c {x l (x, w -- c) >__ ( w, w -- c) }.

From wE bd (C + 8 (C, O) U), c E C and I!c - wll = (c, o) c is the foot of w on C. Thus

(7) C c { x l ( x , w -- c) < (c, w -- c)}.

is no point cEC for which w is the nearest point of O (that is,

foot of c on O), and such that [ ] c - wll = then (5) holds

Suppose now that such a point c exists. Then

it follows that

(6) and (7) imply that C and O are separated by a parallel strip of width

l [ c - wll = Hence there are points of-C such that the nearest point of O is farther away than ~ (C, O), which is impossible. This proves (5).

(8) Let C E e be strictly convex. Then ]or each best approximating polygon

OE ~ o] C all vertices are contained in bd (C + ~ (C, ~.)U) and O

has exactly n vertices.

Obviously Q c C + ~ (C, 8,)U. Suppose not all vertices of Q are contained

in the boundary of C + ~(C, Sn)U. One can obtain from Q a polygon

Q1 E 8, by pushing outwards vertices of Q, if necessary, in such a way that

vert 01 c (C + ~ (C, 8,) U ) \ C ,

at least one vertex of O I (say v) belongs to int(C + ~(C,$n) U),

C c - O l + 8 ( C , ~ , ) U , O l c C + ~ ( C , ~ , ) U .

Let w be the vertex of O 1 next to v in counter clockwise order. If w E int (C + + ~(C, Sn) U) let 0 2 : = O I. Otherwise one can obtain from O 1 a polygon O2ESn by pushing v a little bit outwards, turning the edge conv{v,w}


around one of its points sufficiently near to w and leaving the remaining

vertices unchanged in such a way that

vert 0 2 c (C + ~ (C, 8.) U ) \C ,

at least two consecutive vertices of Q2 belong to int (C + ~ (C, 8.)U),

CcQ2-J -~(C ,~ . )U, Q 2 c C q - 8 ( C , ~ . ) U ,

by (5). Continuing this process a finite number of times one arrives at a

polygon Q"E ~. for which

vert O" c i n t (C + ~ (C, ~.) U) (and thus Q" c i n t (C + ~ (C, 8.) U)),

CCQ"q-~ (C ,~n ) U, O " c C - } - ~ ( C , ~ . ) U ,

By expanding Q" slightly from one of its interior points one obtains a polygon R 'E ~. such that

R c int (C + ~ (C, 8 . ) U),

C c int (R + ~ (C, 8,) U).

Hence ~ (C,R)< ~ (C, 8.). This contradiction shows that all vertices of O belong to bd (C + ~ (C, 8.)U), thus proving the first half of (8). It remains to show that O has precisely n vertices. Consider a set of n boundary points of O containing all vertices of O. By pushing the points which are not vertices of O a little bit outwards one can obtain a polygon S E ~. containing O and having n vertices such that all vertices which are not vertices of Q are contained in int (C + ~ (C, 8.)U). Since this polygon S is also a best approximating polygon of C, all of its vertices have to be boundary points of C + ~ (C, ~ . )U by what was proved already. This is a contradiction unless O has already exactly n vertices. This finishes the proof of (8).

(9) Let C E ~ be strictly convex. Then in each neighbourhood of C there are convex bodies with unique best approximating polygons in ~. .

Let P ES. be a best approximating polygon of C. For 0 < 5 < 1 choose

C (5) E ~ such that

(10) (C, C (5)) = 5 ~ (C, P), 5 (C (E), P) = (1 -- 5) 8 (C, P),

(11) C(5) is strictly convex,

(12) bd(C + ~(C, 8.)U) N bd(C(5) + (1 --5)~(C, 8 . )U)=ver tP .

APPROXIMATION OF CONVEX BODIES BY POI.YTOPES 2 1 9

In order to sce that such a choice is possible note first that by (8)

v c r t P = { v l . . . . . v.} (say) c b d ( C + 8 ( C , ~ . ) U ) .

Let ]~ . . . . . / . denote the feet of v~ . . . . . v,. on C. Now one may take for

C (z) a convex body which is obtaincd by expanding suitably

c o n v ( { ( l - - E ) / , + e v l ] i E { 1 . . . . . n}} U C

so that it becomes strictly convex and so that the points ( 1 - e)fi + ~ vi are precisely its points of maximal distance from C. We show that

(13) for each e the polygon P E ~ . is the unique best approximating polygon

o! C (~).

Let Q E~,, be a best approximating polygon of C(~). By (10)

(14) ~ (C, C (~)) + g (C (E), P) = ~ (C, P) = ~ (C, ~'~).

Hence (4) shows that O is a best approximating polygon also of C. This

together with (14) and the fact that O is a best approximating polygon of

C (e) implies

(C (e), Q) >_ 8 (C, Q) - ~ (C, C (e)) = 8 (C, P) - 8 (C, C (e)) ---

= ~ (C (E), P) > ~ (C (e), Q).

Thus equality holds throughout and therefore by (10)

(15) ~ (C (e), Q) = ~ (C (e), P) = (1 - e) ~ (C, P).

By assumption and (11) the bodies C and C(e) are strictly convex and O is

a best approximating polygon of both of them. Hence (8), (15) and (12) imply

vert O c bd (C + ~ (C, ~d.) U) lq bd (C (E) + (1 - e) ~ (C, b'.) U) = vert P

and O has exactly n vertices. Since P• $. we conclude vert O = vert P and

thus O--= P. This proves (13). Now (9) follows from (13) and (10).

By (3) the sets ~ . are closed. Hence for the proof of the proposition

(16) ~k. is nowhere dense in C /or all k, nE N

it is sufficient to show that ~k,, has empty interior. Suppose that some ~k.

2 2 0 PETER M, GRUBER - PETAR KENDEROV

has non empty interior. By (9) there are convex bodies in (~kn with unique best approximating polygons P E ~',, contrary to the definition of ~k,, thus confirming (16).

Since

t] ~k. = { C E ~ [ C does not have a unique best approximating polygon P E ~. } k=l

we conclude that

~ \ ~ ~ ~k. = {CE ~ [ C has a unique best approximating nffil k=l

polygon P E ~ , for each n}.

Together with (3) and (16) this proves theorem 3 in the Hausdorff metric case.

P R O O F O F T H E O R E M 3 F O R T H E S Y M M E T R I C D I F F E R E N C E M E T R I C C A S E .

In the following, approximation is to be understood in the sense of the metric 0. First we put together some tools for the proof.

The Blaschke selection theorem together with the continuity of Ix and the fact that ~, is closed yields the following:

(17) Let o l , (~2 . . . . 'E Sn be contained in a bounded subset of E d and suppose

that Ix (Ol), Ix(Q2) . . . . > const > 0. Then there is a polygon O E ~,

and a subsequence Oil, Oi2 . . . . such that Oi,, Oi2 . . . . ~ O.

The next proposition is due to Shephard and Webster [27]:

(18) Let Q, 01, 02 . . . . E(~. Then 01, 02 . . . . s__,O implies 01, 02 . . . . O O.

A simple consequence of the proof of Lemma 7 of Eggleston [8], p. 107, is as follows:

(19) Let C E ~ be strictly convex. Then each best approximating polygon

OE ~, of C has precisely n vertices, all o] them exterior to C and

each edge o] O meets the interior of C.

For k, n E N define

qgk,i : = { C E ~ I C has best approximating polygons

Q, R E $ , such that O(O,R)>_ l / k } .


We show first that @k. is c10sed in ~. This will be an immediate consequence

of the following, proposition:

(20) Let C, C I, C 2 .... 6 ~ be such that CI, C2 .... ~ C and let 0 I, O 2 .... E ~'.

be best approximating polygons of C 1, C 2 . . . . . Then there is a subse-

quence Qi,, Qs . . . . converging (in the sense of O) to a best approxi-

mating polygon O s ~. o/ C.

Since C t, C 2 . . . . ~ C and 0 (C t, QZ) = 0 (C l, ~,), O (C 2, O2) = 0 (C 2, ~,) . . . . , the

continuity of 0 implies

(21) 0 (C 1, Q1), 0 (C 1, O 2) . . . . - , 0 (C, ~,).

Obviously 0 (C, 8 , ) < Ix (C). Therefore we may assume that

(22) 0 (C, O~) < ~3, Ix (C) for all i's N

and some constant {3,'6 ]0, 1[. It follows that

(23) ~ (O~) >-- tx (C fl O~) _ (1 - ~1) IX (C) for all i6 N.

We show that there is a constant ~2 > 0 such that

(24) Q i c C + ~ U for all i,6N.

By (23) there is a constant ~3 > 0 such mat for each i there is a circular disk of radius ~3 contained in C fl Q~ c C. Let p 6 O~ have distance [3 from C. Then O i \ C contains a triangle of height ~ and base 2 [33 ~/ ( [3+diamC). The measure of this triangle is less or equal to 0 (C, O~). Thus

_< [3, Ix (C) + diam C

by (22). This proves (24). Because of (23) and (24) it follows from (17)

that there is a polygon Q ' 6 5 , and a subsequence O(,, Q~, . . . . such that

Hence (18) implies

0% Q~, . . . . • Q.

Q~,, Q~, . . . . • Q.

222 P E T E R M . G R U B E R - PETAR KENDEROV

Thus (21) and the continuity of 0 yield O ( C , O ) = 0 ( C , ~ , ) . That is, Q E ~. is a best approximating polygon of C. This finishes the proof of (20). An easy consequence of (20) and the continuity of 0 is that

(25) ~gk. is closed /or all k, n 6 N.

For the proof that ~gk. is nowhere dense in ~ we show first the following elementary result:

(26) Let C, D, R 6 ~ be given such that 0 (C, D) + 0 (D, R) = 0 (C, R). Then C N R c D c C U R .

The inclusions

C \ R c ( C \ D ) U ( D \ R ) ,

imply

R \ C c ( R \ D ) U ( D \ C )

0 (C, R) = I~ ( C \ R ) + is. (R~C) <_

<-- Ix ( C \ D ) + I~ ( D \ R ) + tx ( R \ D ) + Ix ( D \ C ) =

---- 0 (C, D) + 0 (D, R) = 0 (C, R).

Hence equality holds throughout and thus

C \ R = ( C \ D ) U ( D \ R ) , R \ C = ( R ~ D ) U ( D \ C ) .

From this we conclude

(C N R ) \ D = ( C \ D ) O ( R \ D ) c ( C \ R ) N ( R \ C ) = ~ ,

and thus

CARED, D c ( D \ C ) U C c R U C .

Hence the desired inclusions hold.

(27)

D \ C c R

Let C, D 6 ~ and O ~. ~. a best approximating polygon o/ C be given

such that 0 (C, D) + 0 (D, Q) = 0 (C, 0) . Then each best approximating

polygon R ~ ~. o1 D is also a best approximating polygon oJ C and 0 (C, D) + 0 (D, R) ---- 0 (C, R).

The proof of (27) is identical to the proof of the analogous statement (4). Hence it will be omitted.


The essential step in the proof that D~, is nowhere dense is to show the following:

(28) Let C E ~ be strictly convex. Then in each neighbourhood o] C there are convex bodies with unique best approximating polygons in ~,.

Let 0 6 ~, be a best approximating polygon of C. We may assume that o is an interior point of both C and O. Denote the distance functions of C

and Q by q) and ~ respectively (see for example [28], p. 32). For 0 < ~ < 1

the function (1 -- E)q~ + ~ kb is the distance function of a convex body C (0 E ~.

Obviously

(29) 0 (C, C (0) = 0 (s) as E --- O,

(30) C (~) is strictly convex,

(31) C N QcC(OcC U O.

We shall show that

(32) for each ~ the polygon O ~6 ~, is the unique best approximating polygon o / c (~).

Let R 6 ~ , be a best approximating polygon of C(e). By (31)

0 (c , c (~)) + 0 ( c (0 , Q) = 0 (C, O).

Hence an application of (27) yields that R is a best approximating polygon of C and

Thus (26) implies

(33)

0 (C, C (~)) + 0 (C (0, R) = 0 (C, R).

C N R c C ( O c C U R .

According to (19) and the definition of C(E), the differences C ( O \ C and C \ C ( O each consist of n <(lunar~ pieces. Because of (31) and (33) the n pieces of C ( O \ C are contained in O and also in R and the n pieces of C \ C (E) are contained in E d \ O and also in E e \ R . Thus

O N b d C = R N bdC.


By (19),

vert O, vert R c Ed~C, 0 and R each has precisely n vertices and each edge of 0 and R meets intC.

Hence O = R, thus proving (32). From (27) and (32) proposition (28) follows.

(25) shows that ~k~ is closed for all k, nE N. Hence for the proof that

(34) qgk, is nowhere dense in ~ ]or all k, n E N

is is sufficient to show that it has empty interior. But this is a simple consequence of (28) and the definition of Ok,.

Since

oo oo

~ \ U U ~)k. = {CE ~ ] C has a unique best approximating n = l k = l

polygon O 'E Sn for each n},

theorem 3 follows from (25) and (34) in the symmetric difference metric case.

5. Acknowledgement.

The contribution of the first author was written during stays at the Mathematical Institute of the Bulgarian Academy of Sciences in July 1979 and August 1980 and at the Istituto Matematico dell'Universit~ di Bologna in September 1979. For providing these stays we should like to thank Acad. Iliev. and Prof. Papini. We should also like to thank the latter for many stimulating discussions. Many thanks are also due to Prof. Firey and Prof. Groemer.

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Pervenuto il 17 novembre 1980 Institut flit Analysis Technische Universitiit Wien

A-I040 Vienna, Guflhausstr. 27

Institute of Math. and Mech. Bulgar. Acad. Sci.

BG-1090 Sofia, P.O. Box 373

approximation of convex bodies by polytopes

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