expectation of random polytopes

manuscripta math. 91, 393 - 419 (1996) manuscr ipta mathematica O S~nsr ~ 1996

E x p e c t a t i o n of R a n d o m Polytopes

P e t e r M . G r u b e r

Abteilung ffir Analysis, Technische Universit~.t Wien, Wiedner Hauptstrat~e 8 - 10/1142, A-1040 Vienna, Austria e-maih pmgruberQpop.tuwien.ac.at

Received September 12, 1996

Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80 th birthday

Abstract. A random polytope P, in a convex body C is the convex hull of n identically and independently distributed points in C. Its expectation is a convex body in the interior of C. We study the deviation of the expectation of P, from C as n ~ oo : while for C of class C TM, k > 1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodies C of class (71.

MSC 1991. Primary: 52A20, 52A22, 60D05. Secondary: 33D05, 41A60, 53C65, 54E52.

1 In troduct ion

1.1 Let C be a convex body in Euclidean d-space /~ , that is a compact convex subset o f / ~ with non-empty interior. The convex hull of n identically and independently distr ibuted points in C is a random polytope P~ in C. We consider only distributions defined by non-negative, continuous densities on C or on the boundary bd C of C. For surveys of the voluminous literature on random polytopes see e.g. Buchta [4], Schneider [15, 17], and Well and Wieacker [21].

A major problem is to determine the precise value or to study the behaviour as n --. c~ of the expectation of quantities such as the volume or the number of facets of P~. A large number of pertinent results and, in particular, asymptotic

394 P.M. Gmber

formulae are known. Among the contributors are Blaschke, R~nyi and Sulanke, Groemer, Schneider, Buchta, Dalla and Larman, Bs163 and Schfitt.

These results suggest an investigation of the expectat ion of P~ itself. To define it we need the notion of support function hD of a compact convex set D i n L ~ : h D ( u ) = m a x { u ' x : z � 9 D} for u � 9 S a-l, where ''.'' denotes the inner product and S d-1 the Euclidean unit sphere in /E a. Clearly, D -- {x : x �9 u <_ ho(u) for all u �9 Sd-1}. If d(.) is a non-negative, continuous density on C or b d C , then the expectation of P~ is the convex body E~ such tha t hE.(U) is the expectation of hp~(u) for each u �9 S d-~. See [1] and the references cited there.

Bs163 and Vitale [1] proved that in the case of uniform dis t r ibut ion on C there are constants a , fl > 0 with the property that

C [ ~ ] c E ~ c C [ ~ ] for n = l , 2 , . . .

Here C[~] is the floating body of C with parameter r > 0, i.e. the compact convex set which remains if from C all caps of volume e are cut off. Consequently,

En ~ C as n ---* (X:~

where the convergence is with respect to the Hausdorff metric 6 H on the space C of all convex bodies in ~ . 6 g is defined by 6~(C,D) = llhc - hollo~ for C, D E C .

1.2 The aim of this article is to study the deviation of E . from C as n ~ oo, depending on the (differentiability) class C k+l of C (or rather of bd C, considered as a surface).

For C E C N C 1 and x E bd C, let no(x) be the interior normal unit vector of b d C at x. For C E C N C 2 the Gauss curvature o f b d C is denoted by xc . V(.) and A(-) mean Lebesgue measure in ~ , and Lebesgue measure in ~ - 1 or in or surface area measure in ~Y~, respectively.

The notion of affine curvature kc of a convex body in ~ is explained in subsection 2.4 below. It was shown by Klee [10] and the author [5] that for most convex bodies C (in the sense of Baire categories, see 2.5), C C C 1.

In Theorem 1 dis tr ibut ions on C are considered, while Theorem 2 deals with distributions on hd C.

T h e o r e m 1.

(i) Let C G C N Ck+l,k > 2, with Gauss curvature t~c > 0 and uniform distribution on C. Then there are coefficients a2(x), a4(x), a s ( x ) , . . . , ak(x), such that

(1) he(u) - hE,(u) a2(x) n a+ 1

a s n ---~ or

_ _ _ _ ( 1 ) + + . . . + a (f) + 0

n a+"--'f n a+-'--f n ~ n

for x G b d C and u = - n o ( x ) . The constant in 0 ( . ) may be chosen independent of x. a2('), . . . are independent of k and continuous. They can be calculated explicitly. Simple expressions can be given for d = 2 usin 9 ~c and the affine curvature kc of bd C. In particular,

Expectation of random polytopes

(2)

395

a~(x) = F ( g ) A ( C ) ~ c ( z ) ~ 3 ( g ) A ( C ) ~ c ( x ) ~ k c ( x )

12~ , a , ( z ) : (2" 60 ' as(z)= 5F(])A(C)Lw(z)~

9-12~

(ii) Let C C C rq C 2

Then with ~c > 0 and with continuous density de > 0 on C.

d - 1 2 2 (3) he(u) h~o(u)= (d+l)~A(Bd-1)~ \ ~ ] nd,~ / ~ \

for x e b a G and u = - n o ( x ) where o(.) may be chosen independent of x.

(iii) Let (a~), (fl~) be positive real sequences such that

(4) 1 - - o ( a ~ ) a n d / 3 ~ = o ( 1 - ~ _ ~ asn- -*oe . n d~i.1 \ r i d /

Then for most C C C (which may be assumed to be of class C 1) and uniform distribution on C,

~"(c, E~) ~ < ~ ~ for in~nitely many n .

t>Z~ J

A comparision of (1) and (3) yields an explicit expression for the coefficient a~(x).

1.3 In the proof of (i) we use a lemma on the inversion of functions of the form

b,~(w)t m + . . . + bk(w)t k + O(t k+l)

which are strictly increasing in t. The proof of this result is based on the Bfirmann-Lagrange formula for the expansion into Puiseux series of the inverse of analytic functions, see 2.3. A second tool for the proof of (i) is a precise asymptotic expansion of the incomplete beta-function

c~r~

/ ( 1 - t-)~tZdt as n ~ oc, n

0

see 2.4. Integrals of this type are frequently encountered in problems of random polytopes. In the case d = 2 we make use of tools from planar affine differential geometry, see 2.5. Alternatively, the case d = 2 in (i) may be treated in the context of Euclidean geometry. Then the coefficients a2( ' ) , . . , are expressed in terms of ~v and its derivatives, but the formulae are more complicated.

The proof of (ii) is based on ideas which have been applied successfully in the context of approximation of convex bodies by convex polytopes, see reference [8].

Finally, the proof of (iii) makes use of a general result on the irregularity of approximation in the context of Baire categories due to the author [6], see 2.6.

3 9 6 P . M . Gruber

1.4 We draw some consequences of Theorem 1.

1.4.1 It is clear from (i) that for C E CfhC ~176 with xc > 0 and uniform distribution on C, the following asymptotic series expansion holds:

a 2 ( x ) + as( ) + n ho(u)- 7&7. + n--7 '

for x E b d C and u = - h e ( X ) .

1.4.2 For C E C M C ~+1 with ~c > 0 and uniform distribution on C, integration of (1) over all directions u E S g-1 yields the following asymptotic expansion for the difference of the mean widths of C and E,,:

a2 a4 a5 ak ( 1 ~ W ( C ) - W(E,~)= 2 + 4 + 5 + " - + ~ + 0 / / ~ asn---+oo,

n ~ 77, a+'--i" 7Z a+'-"Y n a+-Y \ n a+ x /

where a2, a4 , . . . , ak are suitable constants depending on C. In particular,

2 2 [ . =

a~ = d-, 2 Xc(X)~+=ldO'd-l(x) L (d + 1)a~ A(Sd-1)A(Bd-X)avr b

by (3), where aa-1 = A denotes the surface area measure in /E a. By integrating (3) instead of (1) we obtain for C E C N C 2 with xc > 0 an asymptotic formula for W(C) - W(En) which comprises also the case of non-uniform distribution. Noting that the mean width of En is the expectation of the mean width of P~, these results refine a theorem of Schneider and Wieacker [18].

1.4.3 Given C E CMC 2 with xc > 0, there is a convex body Ca for all sufficiently small ~ > 0, the inner parallel body of C at distance ~5, such that C = Ca + ~SB d (= {x + ay : x E C6,y E Bd}), see e.g. [ll]. Clearly,

hc - hc~ = 6.

Then (ii) yields the following: if the density dc > 0 on C is continuous and 1_

dcI bd C = ~, , then E~ nearly coincides with C6, where

6, = F(a-~'f) 1 d - 1 2 2_.~__ �9

(d + 1)a'+-rA(Bd-1)a-g r n~+,

More precisely,

1.4.4 Again, choose C E CVIC 2 with ~c > 0. Then for all sufficiently small e > 0 the floating body C[d has the property that each of its supporting hyperplanes cuts off from C a cap of volume e, see Leiehtweiss [11]. Thus the expression for the volume of a cap that can be obtained from (54) together with (ii) implies the following: if de > 0 is a continuous density on C with d c l b d C a constant a , say, then E,~ is almost equal to C[~,], where

2 d + l r ( a z ) , 1 ( d + l ) ,

Expectation of random polytopes 397

o r

~SH(E,~,C[~,I)=O ~ a s n - - * ~ .

1.4.5 For C C C Cl C k+l with xc > 0 and V(C) = 1, proposit ion (35) yields the following s ta tement :

for x E bd C and u = -nc (x ) .

1.5 In the following results the vertices of P,~ are chosen on bd G'.

T h e o r e m 2.

(i) Let C E C Cl C k+l, k >__ 2, with xc > 0 and uniform distribution on bd 6'. Then there are coef f ic ien ts b~(~),..., bk(~), such that

h~(u) - hE~ - b~(~)~ + @ + . . . + @ + 0 as n --,. oo rid--1 rid-1 rid-1

for x 6 b d C and u = -no(x) . The constant in 0(.) may be chosen independent of x and b2( - ) , . . . , bk(') are independent of k and continuous. They can be calculated ezplicitly.

(ii) Let C E CfqC 2 with xc > 0 and with continuous density dbac > 0 on b d C . Then

hc(u) - hE,,.(u) = ( d - 1)A(Bd-a) d@r \ d ~ 2] n ~-~

for x e bd C, u = -no(x ) and o(.) may be chosen independent of x.

(iii) Let (c~), (~,~) be positive real sequences such that

m - o ( ~ ) an d ~ . = o as n --* ~ . nd-X

Then for most C e C (of class C ~ ) and uniform distribution on bd C,

{ <> c~, } for infinitely many n. 6H(c, E,.,) ~,

Since the proof of Theorem 2 is similar to that of Theorem 1, we offer an outline. Clearly, we may draw consequences from Theorem 2 which are similar to those drawn from Theorem 1.

1.6 In the following bd, relbd, int and conv shall s tand for boundary, boundary relative to the affine hull, interior, and convex hull.

398 P.M. Gruber

2 P r e l i m i n a r i e s

2.1 D e f i n i t i o n s a n d N o t a t i o n . Let C E C. A supporting hyperplane of C is a hyperplane which meets bd C, but not

intC. For each u E S d-1 there is a unique supporting hyperplane He(u) of C with normal vector u, such that u points into the halfspace bounded by He(u) and not containing C. The cap Co(u, h) of C with normal vector u E S d-1 and height h > 0 is the par t of C between He(u) and He(u) - hu. Its base Be(u , h) is the set C M ( H c ( u ) - ha). Now assume that C 6 C 1. Let x 6 b d C and u = - n c ( x ) . We then write Hc(x) , Cc(x, h) and Bc(x , h) for g c ( u ) , Cc(u, h) and Bc(u, h ), respectively.

The width we(u) of C in direction u E S d-~ is hc(u) + h c ( - u ) . For notat ion not explained below and for s tandard results and arguments we

refer to Schneider [16] without further mention�9

2.2 A r e m a r k on s m o o t h c o n v e x b o d i e s . A straightforward extension of an argument of Schneider [14] from k = 2 to general k provides our first tool (the reader is warned tha t this result is not trivial).

L e m m a 1. Let C E C M Ck+l: k __> 2, with •c > O. Then there are constants a, fl > 0 such that the following hold: let x E bd C and choose in Hc(x ) a Carte- sian coordinate system with origin x. Together with he(X) it forms a Cartesian coordinate system for JE d. Then the open a-neighbourhood No(x , a) of x in bd C can be represented in the form

( u , f ( u ) ) : u e

Here " ' " denotes the orthogonal projection of ~ d into He(x ) and f is a convex function of class C ~+1 for which the absolute values of the partial derivatives of f up to order k are bounded by/3.

2.3 I n v e r s i o n of f u n c t i o n s . We prove the following result.

L e m m a 2. Let

z = z(w, t) = bm(w)t TM + . . . + bk(w)t ~ + O(t k+l) for 0 < t < c~,

2 < m < k, be a strictly increasing function in t for each fixed w in a given set. Assume that b,~(.) is bounded between positive constants, that b,~+l( - ) , . . . , bk(') are bounded~ and that the constant in 0( . ) may be chosen independent of w. Then there are coefficients c1( ' ) , . . . , Ck-m+l('), and a constant 7 > 0 independent of w, such that for each fixed w the inverse function t (w, . ) of z(w, .) has the representation

t ----- t ( W , Z ) = C l ( W ) Z 1 "3L. . . + C k _ m + l ( W ) Z rn "~- O(z k - ~ 2 ) fovO < z < 7.

The coefficients c1(. ), . . . , ck-,,,+l(') can be determined explicitly in terms of b,~(. ), �9 . . , bk('); in particular,

1 b,~+l(')

c3(') - (m + 3)bm+l(.) m b m ( . ) ~ + 2 2,,+3 , 2m .


b,~+a(') (m + 4)b,,,+x (')b~+2(') (m + 2)(m + 4)b,~+, (.) 3 c 4 ( . ) = m b ~ ( . ) ~ + ~ ~ + , - ~ ~+, m b.~( . ) ~ 3 m b ~ ( . ) -, The coe f f i c i en t s are b o u n d e d a n d i f bm( ' ) , . . . ,bk( ' ) are c o n t i n u o u s , so are

c t ( - ) , . . . , ck-,~+l(') a n d the c o n s t a n t in 0 ( . ) m a y be c h o s e n i n d e p e n d e n t o f w .

Proof . We need the following result on the inversion of analytic functions, see e.g. Hurwitz-Courant [9], p.139f, Markushevich [12], p.92f, or Vinogradov et al. [20], col.5, p.183.

(5) Let bin, b,~+l,. . . E /R, b m > 0, m > 1, be such that the series

z = z ( t ) = brat m + b,~+lt m+~ + . . .

is convergent and its sum z(.) is strictly increasing for all sufficiently small t > 0. Let

1 ~ t ~ / ( ' - ' ) ct = - lim ~ for l = 1 ,2 , . . .

t! , -+o [ z(t)-~ J

Then the series 1 "2

t = t ( z ) = c l z ~ + c 2 z ~ + . . .

is convergent and its sum t(-) is the inverse function of z(-) for all sufficiently small z >_ 0.

Clearly, by the general binomial theorem,

( 6 ) t t

~( t )-~

bm+~ t bin+2 t2 + . . . _ t l b ~ t - t { 1 + T } - - ~ where T = bm + b m

'

= bm~{1 + T + T 2 + . . . }

= b::S l _ L T + 1 L ( L �9 + 1 ) T 2 - + .) ?'rt -2 lTt m ""

t I b ~ i t . 4 _ ( - - ' 3 L b7~2+ ) = b ~ { 1 l bin+2 1 l l + m 1 t 2 + . . . } . rn m bm 2 m m

(5) then shows that

(7) ct is the coefficient of t t-1 in (6) divided by l. Hence it can be determined explicitly; it is a polynomial in b,~, . . . , bin+l-1 divided by a power of bin. In particular,

1 b,~+l b.~+2 (m + 3)b2m+l Cl ~ - " ~ , C2 ~ ~ , C3 ~ vn+3 "Jl- 2rn-~-3

b~ mbm ~ mbm ~ 2m2bm ~

3 b~+3 (m + 4)b~+,bm+~ (m + 2)(m + 4)b~+1 C4 -- ~ + ~,,,+4 -- z.~+4

mb,~ ~ m2bm m 3m3bm m

400 P.M. Gmber

After these preparat ions the proof of Lemma 2 is as follows. By the assumptions on the coefficients b,~(.) , . . . , bk(.) and on O(.) there is a constant bk+l > 0 such that for the functions

z• = b,~(w)t '~ + . . . + bk(w)t k :l= bk+lt k+l

the following holds, where we have replaced the a in the assumptions of Lemma 2 by a smaller constant, again denoted by a, if necessary.

(8) z - (w , t ) < z(w, t ) < z+(w,t) and z(w, t ) ,z• all are strictly increasing in t for each fixed w and 0 < t < a.

By assumption,

s u p " b'~+l(w) ', ~ = " b~(w) . . . , I I,I t: w} < + ~ .

Since by (8) for each fixed w the functions z*(w, t) are strictly increasing for 0 < t < c~, an application of (5) and (7) shows that

(9) for each fixed w the inverse functions t• .) of the functions z• may be represented in the form

t • = Cx(W)Z-~ + . . . + ck_m+ltw~z ~ + ck_ ,~+2twjz ~ + . . .

for all sufficiently small z > 0 with suitable coefficients ct(w), c~(w).

Next, some properties of these coeff• will be given. First,

(10) c1 ( ' ) , . . . , ck-m+l( ' ) can be determined explicitly, in particular,

1 bin+l(') c~(.)_ ~(.)_~, c~(.)=- mb~(.)~--~'

bm+~(-) (m + 3)b.,+~(.) ~ c ~ ( . ) - mb~(.)~--~ ~ + 2m~b~(.)~'+~ '

bin+3(') (rrt + 4)bm+l(.)bm+2(') (rrt -I- 2)(m -4- 4)bin+l(') 3 3 3m~-4 , c4(.) m b , ~ ( . ) ~ + rn2b~(.)2~ +' 3m b,~(.) ,~

and they are bounded.

This follows from (7) and the assumptions of Lemma 2. Second,

(11) if bm(') , . . . ,bk( ') are continuous in w, then c1( ' ) , . . . ,Ck-m+l( ' ) are also continuous in w.

Again, this is an immediate consequence of (7) and the assumptions of Lemma 2. Third,

(12) there is a constant ~ > 0, independent of w, such that Ic~(w)l < fit for e a c h w a n d l = k - m + 2 , . . .


The proof of (12) is more complicated: if in (5) b,~,bm+h.., are replaced by b,~(w),. . . , bk(w), +bk+~,0, 0 , . . . , we see from (7) that ct(w), rasp. c~(w) i s the coefficient of t 1-1 in the power series in (6) divided by I. In the present case T is a polynomial in t having at most (k + 1 - (m - 1) _<)k coefficients which, possibly, are :# 0. The absolute values of these coefficients all are < a by the definitions of T and a. Clearly, T i is a sum of at most k i products of i monomials from T. Thus each coefficient of a power of t in the polynomial T i has absolute value < kic ri for i = 1 ,2 , . . . This gives the following very rough estimate for tc,~(w)l,1 = k - m + 2 , . . .

1 t ( t 1 { t k ~ + ~.-~ -~ + 1)k~ + . . . <_ lb~

1 ! (2+1) (s - - . . . + l - 2)k~-la 1-1} " " + ( l - 1 ) ! m m m

l(t + 1)k~o.~ t ( t+ l ) . . . (2 t 1 { l + I k a + + . . . + - 2 ) k t - l a l - 1 } < '-- 2! ( 1 - 1 ) ! lb~

_< 1-~{1_ + k a + k2a2 + " " + 21 _z , lb~ lb~

concluding the proof of (12). By (9), (12) and the sum formula for the geometric series we see that

(13) for each fixed w the inverse functions t+(w, .) of z• .) may be represented in the form

t+(w,z) = c,(w)z~ + . . . + c~_~+ltw)z ~ + o~(z %-=~)

for 0 < z < 2P--~, say, where the constant in 0• may be chosen independent of w.

The assumptions on bin( ' ) , . . . , bk(') in Lemma 2 show that - - after replacing a by a smaller positive constant independent of w, if necessary, again denoted by a - - we may assume that

(14) there is a constant 7 > O, independent of w, such that

0 < -~ _< z ~ (w, ~) < ~- -

for each w.

By (8) and (14) the inverse functions t(w,-) and t+(w, .) of z(w, .) and z:~(w, .), respectively, satisfy

(15) t+(w,z) <_ t (w , z )<_t - (w , z ) for each f i x e d w a n d O < z < 7 .

Propositions (13) - (15) together show that there is a constant 7 > O, independent of w, such that we get the desired representation of t(w, z) for each fixed w and 0 < z < 7, where the constant in 0( .) may be chosen independent of

402 P.M. Gmber

w. The properties of the coefficients C 1 ( ' ) , . . . , C k _ m + l ( ' ) asserted in Lemma 2 are immediate consequences of (10) and (11). This concludes the proof of Lemma 2.

2.4 On t h e i n c o m p l e t e b e t a - f u n c t i o n . The following result is a refinement of a formula of SchlSmilch, see Whittaker and Watson [22], p.242.

L e m m a 3. Let 1~ E IR. Then there are coefficients bl, b2,... E /R depending on 13 which can be determined explicitly, such that for fixed l = 1 ,2 , . . . and 0 < a < 1

1 - ' ~ t ~ d t = F ( f l + l ) + - - + . + + O a s n ~ c c . n "" ~-l

o

In particular,

b, F ( ~ + 3 ) b~ F( /3+4) F ( ~ + 5 ) 2 3 8

I r a is chosen from a closed subinterval of (0,1], then the constant in 0( . ) may be chosen independent of a.

Proof. Let 0 < ~- < 1'. Then r~

(16) { e ~ ( 1 - ' '~ ~)} = {1 - T} ~

where

t 2 It~+l t 2 ltt+ 1 ( t l+~

T = ~ + . . . + ( l + 1)!nt+l + . . . . 2n 2 + " " + ( l + 1)In'+1 + O \n-7~ ] _< 1.

Thus Taylor 's theorem shows that for suitable ~ E (0, 1),

(17) {1 - T } ~

('~n 2 ...-I--(-1)' (nl)TCt-(-1)'+'(l:ll(1-v~T)"-'-'T'+' k - - / \ - - /

= 1 - n2~ ~ . . . + o k-r~r]

+ 4n---Z + ' ' + O \ - ~ ]

+(-1) t 2- r~,+. . .+Okn,+, /+Okn-~ /

p,(t) + 0 ( t'+~ + e'+~'~ = l + p ' ( t ) ~ + . . . + st \ ~z++ },

where P l ( ' ) , . . . ,Pl( ') are polynomials in t which may be calculated explicitly. In particular,

Expectation of random polytopes t 2 t 3 t 4

(18) pl(t) = - - ~ , p2(t) = ---ff + g.

(16) - (18) together with the definition of the F-function yield,

a n a n

/ ( 1 - t ) ' ~ t O d t = / ' ' t e- {e~(l- )}"t'at n

o o c~n

= / e_t{ 1 + p,(t)n +" "" + ___~i_+O / t t+2~+~ • t 2t+2\)}t,dt 0 +oo +e~

= / Wdt- / W,tt 0 Otn

+oo b~ =b~ ) + ~ / ~-�89 )It~

0

= bo + _bin + "" + ~7+obt (hi_ ~ ) ,

403

where + o o + o o

bo = / e-tt~dt -- F(/3 + 1), bl = / e-tp,(t)tt~dt - F(/3 + 3) 2 '

0 0

+ o o

b2= F ( ~ 3 ~ + ~ " + 4 ) F(~3+5) . - , b , = _/ e-tpl(t)tVdt. 0

The proof of Lemma 3 is complete.

2.5 Tools f r o m p l a n e aff ine d i f fe ren t ia l g e o m e t r y . For the following compare e.g. Blaschke [2] or Simon [19].

Let C be a plane convex body of class C ~+1, k >_ 3, with curvature xo > 0. Let t denote the arclength of bd C, measured in counter clockwise direction. The affine arclength s of bd C is defined by

d 1 (19) ~ a ( t ) = xc(t)~

and differentiation with respect to s is denoted by " ' " Let xc be a corresponding parametr izat ion of bd C. Then

(20) tc=x~,mc=x~ are the affine tangent and normal vector of bd C. Then

(21) det(tc, mr 1

and ke = det (mc, m~)

is the afflne curvature of bd C. The formulae

' -kctc t ic = rnc~ m C =

yield the following form of the Taylor expansion for xc and z~ at s = 0.

4O4 P. M. Gruber

(22)

(23)

= c ( s ) = = c + (s ko(o) ,3 kb(o) s, + ko(o) 2 - k'b(O)ss - ~ 24 120 +.. .) tc(O)

+ (-~s 1 2 kc(O)s424 kc(O)ss+60 ...)rnc(O),

=~(~) = (a - kc(~ k~'(~ + kc(~ - vb(~ ---7- 6 24 +...)to(o)

+ ( s - k~176 kb(~ .)too(o). 6 12 ""

These notions all are invariant, resp. equivariant with respect to area preserving affinities.

2.6 Tools f r o m B a i r e ca tegor i e s . A topological space is Baire if any of its meager subsets has a dense complement, where a set is meager or of first Baire category if it is a countable union of nowhere dense sets. When speaking of most elements of a Baire space we mean all elements with a meager set of exceptions. A version of Baire's category theorem says that any locally compact space is Baire. Since the space C of all convex bodies, endowed as'we assume with the topology induced by the metric 6 H, is locally compact by Blaschke's selection theorem, we may speak of most convex bodies. See e.g. Gruber [7]. A special case of a result on the irregularity of approximation in [6] may be stated as follows.

L e m m a 4. Let X be a Baire space, (~o~) a sequence of non-negative real continuous functions on X, and (a,,) and (•) positive real sequences. Assume that both the sets

{= e x : ~ . ( = ) = o (~ . ) . s ~ -+ oo}, {x e x : 8 . = o (~ . (=) ) as n --* o~}

are dense in X . Then for most x 6 X ,

p,~(x) < for infinitely many n. > ft.

3 P r o o f o f T h e o r e m 1

3.1 A n exp l i c i t r e p r e s e n t a t i o n o f hen . Let C 6 C and let dc be a continuous density on C. Then

~c(~)

hv(u) - hE,(U) = / (1 -- V(u, h)l"dh 0

where V(u,h) = / dc(y)dy.

Cc(u,h) I[ dc = V(C) -1, then clearly

V(u, h) = V(Cc(,,, h)) v ( c )

for u 6 S ~-1,

Expectation of random polytopcs 405

Compare [11. In case C e C a and x e b d C , u = - n c ( x ) , we write for V ( u , h ) also V(z, h). 3.2 P r o o f o f (i)

3.2.1 In this subsection a precise asymptotic formula for the volume of a cap of C will be given:

(24) There is a constant 6 > 0 such that the following holds. Let x E bd C. Then there are coefficients bd+l(x), b~+3(x),bd+4(x),. . . , bd+k-l(z) , such that

v(cc(x , h)) = b~+,(x)h~ + b~+~(z)h~ + b~+~(z)h~ + . . .

�9 . . + b ~ + k - l ( z ) h 2 + O ( h ~ ) f o r 0 < h < s

bd+l(') > 0 and bd+a( ' ) , . . . ,ba+k-l( ' ) are continuous in z and can be calculated explicitly. The constant in O(.) may be chosen independent of X .

The assumptions C E C k+l and x c > 0 in (i), Lemma 1, and Taylor 's theorem imply the following propositions:

(25) There is a constant 0 > 0 such that for each x E bd C the project ion N c ( x , ~) ' of N o ( x , ~) into H e ( x ) contains a circular disc of radius ~ and center x.

(26) Let x E b d C , v G S d-2, and 0 < r < ~. b2(x, v), . . . , bk(x, v) , such that

Then there are coefficients

z = f ( r v ) = b2(x ,v )r 2 + . . . + bk (x , v ) r k + O(r~+l).

b2( ' , ' ) is bounded below by a positive constant independent of x , v . In addit ion, b2(., . ) , . . . , bk(', ") are bounded, they are continuous in v for fixed x, and can be determined explicitly. The constant in O(.) may be chosen independent of x, v.

Since z is a s tr ict ly increasing function of r for 0 < r < ~o, proposit ion (26) together with Lemma 2 gives the following,

(27) There is a constant a > 0, such that for all x E bd C , v E S d-2 there are coeff• c~(x, v ) , . . . , c~- l ( z , v) with the property that

r = c , ( x , v ) z � 8 9 + . . . + c ~ _ , ( x , v ) z ~ + O ( z } ) for 0 < z < c,.

The coefficients can be determined explicitly in terms of b 2 ( ' , ' ) , . . . , bk( ' , ' ) . In part icular ,

�9 1 b3(x, v) el(z,v) b~(x,v)~' c~(x, ,)- 2b~(x,~)~"

They are bounded, for each fixed x continuous in v, and the constant in O(.) may be chosen independent of x , v .

406 P.M. Gruber

Raising r in (27) to the (d - 1)th power, yields the following proposition:

(28) Let x E bd C, v E S d-x. Then there are coefficients d~(x, v ) , . . . , dk_~(x, v) such that

~ / d + k - - 2 ~

r d-1 =dl(X,V)Z ~ + . . .+dk_ l (X ,V)Z " ~ ~ + t ) t z ~ ) for 0 < z < a.

The coefficients can be determined explicit ly in terms of the coefficients c1(', "),. �9 Ck-1 (', "), in part icular,

dl(x, v) = cl(x, v) d-x, d2(x, v) = (d - 1)cl(x, v)d-2c2(x, v).

They are bounded, for each fixed x continuous in v, and the constant in O(.) may be chosen independent of x, v.

For x E bd C, v E S d-2' and z = f (rv) we have represented r d-1 in terms of z, see (28). Thus we can calculate first the area of Be(x , z) = O A (Hc(x) - zu), where u = - n e ( x ) , and then, by integrating from 0 to h, the volume of the cap Co(x, h). O'd-2 is the (d - 2)-dimensional surface area measure on S d-2.

(29) Let x E bd C. Then there are coefficients bd+l(x),. . . , bd+k-~(x) such tha t

h h

0 0 S a -2

= bd+l (x )h~ + . . . + bd+k_l(x)h ~ f z ! + O ( h ~ ) for 0 < h < ~r.

The coefficients are obtained from dx(x, "),... ,dk-x(x, ") by integrat ing over S d-2. They are bounded. The constant in O(.) may be chosen independent of x.

We now investigate propert ies of bd+ l ( ' ) , . . . , bd+k-~ ('). Firs t ,

(30) bd+2(x) = 0 for each x E bd C.

To see this, note that for given x E b d C the positive quadra t ic form b2(x, .) in (26) is even and the cubic form b3(x, .) is odd. Thus the coefficient cl(x, .) in (27) is even and c2(x,.) is odd. This in turn implies that the coefficient d2(z,.) in (28) is odd. Hence ba+2(x) = 0 since it is obtained from d2(x, .) by integrating over S d-2.

Second,

(31) ba+l ( '), bd+3('), �9 �9 �9 bd+k-1 (') are continuous.

Assume the contrary and let bt(.) be the first of these coefficients which is not continuous. Since bd+ l ( ' ) , . . . , b l - l ( ' ) are continuous and bt(-) is bounded (see (29) for the la t ter) , there are x0, X l , . . . E bd C with xi ~ x0 as i ~ (x~ and

(32) b~(x,) ~ b,~(Xo) for m = d + 1 , . . . , 1 - 1,

bt(xi) ~ bl, say, where bt ~ bt(xo).

Expectation of random pol.vtopcs 407

An elementary argument shows that V(x, h) = V(Cc(x, h)) is continuous in x, h. Hence

(33) Y(x, , h) ~ V(xo, h) for each h, 0 < h _< a.

By (29),

V(xi, h) - V(x0, h) - (bd+l(xi) - bd+l(xo))h~ . . . (bt(xi) - b~(xo))h~ - . . .

. . . . - (ba+k-l(xi) - b d + k - l ( x o ) ) h 2 O(h 2 ) for 0 < h < a,

where the constant in O( ' ) may be chosen independent of xi, xo. Letting i --~ r and noting (29), (32) and (33) we arrive at

- (b , -b , (xo) )h �89 +O(h 2 ) = O ( h 2 ) f o r 0 < h < a .

Since bl ~ bl(xo) and l + 1 < d A- k, this is impossible, concluding the proof of (31).

Third,

(34) 5d+l(') > 0.

For fixed z the coefficient b2(x,v) in (26) is a positive quadratic form in v (note that ~c > 0). Hence Cx(X,v)in (27) and d , (x ,v ) in (28) are positive for v # 0 and (29) yields (34).

Having shown (29), (30), (31) and (34), the proof of (24) is complete.

3.2.2 Here we express the height h of a cap Co(x, h) in terms of the volume of the cap.

(35) There is a constant r > 0 such that the following holds: given x E bd C, u = - n o ( x ) , there are coefficients c2(z), c4(x), cs (x ) , . . . , ck(x) such that for

V(Cc(x , h))~ ,~ : v (~ , h) ( : v(~, h ) - - Y ~ "

we have

h h(z, y--) Y ~ Y '-- = ,~ = c2(x)(X)~§ + c4(~)(X)d+, + . . .

Y "'" + ck(X)(n)~+~ + o((Y-) dk+--~ ) n for 0 < yn < ~'"

C2(') > 0 and c @ ) , . . . , ck(') are continuous in x and, in principle, can be calculated explicitly. The constant in O(.) may be chosen independent of x. h is continuously differentiable as a function of y and has positive derivative.

By (29) V(x , . ) and thus y is continuously differentiable in h with positive derivative. Hence the inverse function h also is continuously differentiable in y and has positive derivative. We may apply Lemma 2 to (24) to express h�89 in terms of y. Squaring then gives (35).

3.2.3 In this subsection he(u) - hE,(n) will be calculated. We first refer to (35) and show that

408 P.M. Gruber

(36) there is a constant r / > 0 (depending on r ) such that

h(x ,r )>r/ for a l l x E b d C .

h(x, r) is the height of a cap of C of volume TV(C). Since C is strictly convex, (36) follows.

Second, we show the following.

(37) Let x 6 b d C , u = -ne(x) . Then there are coefficients a2(x), a4(x), a s ( x ) , . . . , a~(x), such that

h(x,~-) f (1 - V(u, h))"dh - a2(x)2

n-X~-f o

_ ( 1 ) _ _ _ + a4( ) + . . , + a k ( ) + 0 ~ .

n +~%-T n a--4-f n k+;r~

a2( - ) , . . . , ak(-) are continuous and in principle can be calculated explicitly. The constant in O(.) may be chosen independent of x (or u).

By (35),

h(x,~) ~,~

f f ( Y.. "h'(Y-)dYn l (1 - V(u, h))"dh = 1 - -~) n

o o

2C2(X) ~'/ 1 ~ rf Y,n a~y-1- 1 d + 1 (1 - Y - - ) n y d - ~ - f - l d y " 2 q- q- (1 - -

- - - . . . n ) y + a y k n n ~TY -I- ?2 d+---Y

o o Tn

0 n d+1

where c2('), c4(-), c5 ( ' ) , . . . , ck(') are continuous and in principle can be calculated explicitly. The constant in O(.) may be chosen independent of x. Now, calculat- ing the integrals according to Lemma 3 suitably and collecting the O(-) terms, gives (37).

Third, by (36),

(38) ~c(~)

I / (1 - V(u,h))'~dhl < (1 - V(u,h(x,T)))'~(wc(u) - h( x, "r ) ) h(~,~)

< (1 - V(u, rj))'~wc(u) = 0 ~ , /2

where the constant in O(.) may be chosen independent of z (or u).

Finally, the representation of hE, in 3.1, and propositions (37) and (38) yield (i), except for the assertion in case d = 2.

3.2.4 In the last par t of the proof of (i) we describe a method to determine for d = 2 the coefficients a2(x),... Since tedious calculations are required, only a2(x), a4(z) and as(x) will be derived explicitly�9


Let x E b d C . Choose an affne arclength parametrizat ion x(s) with x = x(0). Our a im is to calculate areas. Hence we may apply an area-preserving affinity (and change notat ion) such that to(O) is orthogonal to me(O) without

changing the s i tuat ion essentially. Then (19) - (21) show that ~c(O)}tc(O) and 1 ~c(O)-~mc(O) are the Euclidean tangent and normal unit vector of C at x.

Thus in the Car tes ian coordinate system with origin o and these vectors as basis vectors, the affine arclength parametr izat ion x(s) = (~(s), r/(s)) has the form

, kc(0)s3 k~) s, O(sh)), (39) ~(s) = ~c(O)-~(s- T - +

i ~ k~(O)s, kb(O)r rl(s ) = ~o(O)~(~s 24 - 60

and, in addit ion, we have

(40) ~'(s) = tcc(0)-�89 - kc(O)s2 - - - 2

kb~~ + O(s'))

k~(O)s~ kb(O) s, + o(r ~'(s) = ~(o)~(s - - - C - 12

as s ---* 0, see (22) and (23). First, the following will be shown.

kb(0) s ' + O(s s) (41) Let s , t > 0, such that r/(s) = r / ( - t ) . Then t = s - - - ~

a S S ----~ 0 .

By (39)

1 t 2 /w(0) z = ~c(O)-~rl(s) - 2 24 t ' + kc(O)ts + O(t6).

6O

An applicat ion of Lemma 2 then shows that

' v ~ k c ( ~ - k b ( ~ (42) t = v~z~ + 1 ~ 15

Noting that

z = ~-(1 s2 - kc(O)s212 k~ + O(s4))

1 3_ 2 5_ by the assumption in (41), we may express z~, z2, z , z2 in terms of s by means of the general binomial theorem. Inserting the expressions thus obtained into (42) then yields (41).

Second, we de termine the area of a cap in terms of s.

(43) For s > 0, let A(s) be the area of the cap of C with basis equal to the chord x(s), x ( - t ) , see (41). Then

A(~) = ~s2 ~ 2ko(0)r k~(0)r + ~ as s --, 0.

410

Clearly, $

A ( ~ ) = (~(~) - ~ ( - t ) ) ~ ( ~ ) - / rl(r)~'(r)dr. - - t

Hence, using (39), (40) and (41), proposition (43) follows. Third, we calculate the height of a cap in terms of its area.

P. M. Gruber

(44) For A > 0, let r/(A) be the height of the cap of area A with basis parallel to the first coordinate axis. Then

1 3 2 , ~ , 3 , ~ 3 k c ( 0 ) A~ rl(A) = a c ( 0 ) ~ { ( ~ ) ~ A ~ + (~) ~ + O(A2)) as A ---, 0

and 7/is differentiable for A positive and sufficiently small.

In (43) we have expressed the area A of the cap of height r/(s) in terms of s. Clearly, A is strictly increasing for s > 0 and s sufficiently small. Hence an application of Lemma 2 implies,

3 , . , kc(O)A (3-)~k'c(O)A~+O(A[). = (-~)~A~ + -7-6-- + 2 40

Inserting this into the formula for q in (39), gives the desired expression for r/. A is differentiable in terms of r/, the derivative being the length of the chord which is the basis of the corresponding cap, compare (29). Hence the inverse function 77 is differentiable in terms of A for A > 0, concluding the proof of (44).

Fourth, let u = - n c ( x = o). Then by 3.1,

(45) he(u) - hE.(u) =

Putt ing

~c(~) A(rl) ,~ �9

f ( 1 - A----~) ar/. 0

y A(r/)

n - A ( C )

and using (44), the integral in (45) is transformed into a sum of integrals of the type considered in Lemma 3, the variable now being y. An application of Lemma 3 finally gives the desired expressions for a2(x),a4(x),ah(x). This concludes the proof of part (i) of Theorem 1.

3.3 P r o o f of (ii) Let 1 < )~ < 2 be chosen.

3.3.1 First, some preparations are needed. For p E bd C choose in Hc(p) a Carte- sian coordinate system with origin p. Together with nc(p) it forms a Cartesian coordinate system of/U/. Let " ' " denote the orthogonal projection o f / ~ onto He(p) and choose an open neighbourhood N(p) of p in bd C such that g(p)' is an open (d - 1)-bali in He(p) , the closure of which is contained in the interior of C ' relative to He(p). If N(p) is represented in the form

x = (u, f ( u ) ) : = z N ( p ) ' ,


then, since C ECf3C 2, the function f is convex and of class C 2. For each u E N(p)' define a quadratic form %,~ by

1 %,~(s)= ~ f , i j ( u ) s i s j for s----(sX,. . . ,s d-1) 6 ~d-1,

where i , j = 1 , . . . , d - 1, and the f,ij are second partial derivatives, xc > 0 and f 6 C 2 imply that %,~ is positive and its coefficients are continuous in u. By decreasing N(p) if necessary, we thus may assume that

lqp,p(s) < %,~(s) < A%,v(S ) u E N(p)', s E for ~-1,

and, in addition,

(1 + y~ f , , (u)2) ~ _< A for u C N(p)' , i

1 ~x~c(p) < xc(x) <_ A~o(p) for x E N(p),

1 -~dc(p) <_ de(x) < Adz(p) for p C cony N(p).

Here i = 1 , . . . , d - 1, and the f,i are first partial derivatives. As p ranges over bd C, the corresponding open neighbourhoods N(p) cover

the compact surface b d C . Hence there are points pz 6 bdC, l = 1, . . . ,m, say, and corresponding open neighbourhoods Nt which cover bd C, convex functions ft G C 2 and positive quadratic forms ql = qw,v. and q= -- %.,, having the following properties.

1 (46) -~ql(s) <_ q,,(s) < Aqt(s) for u e N[, s e ~d-, ,

(47) (1 + ~ f t , i 2 ~- (u ) )2 < A for u C IV[, i

1 (48) i x c ( p , ) < ~c(x) _< Axz(p,) for x e N,,

1 (49) ~dc(pt) <_ dc(x) <_ Adc(p,) for z 6 conv Nt.

Clearly, we may choose the Cartesian coordinate system in Hc(p) such that

~,(p,) ~ ~ ~ - (P,)(~d-1)2 (50) qt(s)= 2 ( s ) + . . . + - - ~ for s C L a - ' ,

where x~(Pt) . . . . . tcd-~(pt) are the principal curvatures of bd C at pl. The open neighbourhoods N1 , . . . , N,~ cover bd C. Thus Lebesgue's covering

theorem together with the strict convexity of bd C (note that ne > 0) shows that

(51) there is an r > 0 such that for any x 6 bd the cap Cc(z,e) is contained in cony Nt for a suitable I.

412 P.M. Gruber

We note that e depends on .~.

3.3.2 In order to determine A(Be(x, h)) and V(Cc(x, h)) we first show the following.

(52) Let x = (u, fi(u)) e Nl. Then

{ <- ~1 }.qt(s) for u ' , u + s e N [ . ft(u § s)-- f t ( u ) - ~ > -~

This is an immediate consequence of the convexity of N[, Taylor 's theorem, the fact that ft E C 2 (which follows from the assumption that C E C 2) and (46):

1 f t ( u + s ) - f , ( u ) - ~ f , , i ( u ) s i = ~ y ~ . f t , i j ( u + O s ) s i s j - ~1 "qt(s),

i i,j > -~

where 0 < 0 < 1 is suitably chosen (depending on a). Choose e > 0 such that (51) holds. Next the following will be shown.

(53) Let x E bd C. Then

1 . 2 - r ' A - - ~ ( B ~ - ) h ~ for 0 < h < e . A(Bc(x, h)) >- A--a~-2= nc(z)~

By (51) there is an I such that relbd Be(x, h) C Nt. Hence we may represent relbd Be(x, h) as follows, where x = (u, ft(u)):

relbd Be(x, h) = {(u + s, ft(u + s ) ) : ft(u + s) - ft(u) - ~ ft,i (u)s i i

2 1 = h(1 + ~ f t , i ( u ) )7}.

i

Thus Bc(x,h)' = { u + s : f , ( , , + s ) - i , ( u ) - ~ f , , , ( , , ) 2 ) ~ }

i

_ { u + s : q ( s ) > ~

by (52) and (47). Combining this with (47), (50) and (48), we see that

< ~ ( p , ) . - . ~ ~A(e ~-~) A(Bc(x, h)) 1 2h 2h ~_ d-1

>- -~(Ax,(pt)... A~d-7(p,))~d( B )

~d+l ] a-x d 1 2-r-A(B - ) ,- , 1 / �9 ~ h-r-, _< >

concluding the proof of (53). Finally, integrating (53) yields the following.

Expectation of random polytopcs

(54) Let x E bd C. Then

v(co(x, h)) > , ~ d + l / 2da-~OA(Bd-1) h d2"~-!"

1 �9 - - _ .------7-~. ~ A---~ J (d + 1)~c(x)~

413

for O < h < e.

3.3 .3 We are now ready to prove (ii).

(55) Let x E bd C, u = -nc(x). Then (3) holds, where o(-) m a y be chosen independen t of x.

We refer to 3.1�9 Choose r > 0 and l according to (51). T h e n (49) and (54) together yield the following�9

{ <- ~1 }'dc(pl)V(Cc(x,h~) (56) V(x,h) > A

{ < Ad+3 } 2~A(B d-l) dc(x~,d_.~ 1 �9 �9 - - t z 2 for 0 < h < e .

> - - - ~ d + 1 ~ o ( x ) -

By chosing e > 0 smal ler , if necessary, but still independent of x, and r enaming it, we m a y assume tha t

(57) e < min{wc(u) : u E S d-l}

and tha t the two expressions on the right hand side in (56) b o t h are less t han 1 for h = g. T h e n by (56),

. . . . h=)dh . o - 0 Ad+3 d + 1 no(x)�89

Subst i tu t ing

t A'-7~'~ . 2d~~~ d-l) . de(x) . d_ .+~

n = /~d+3 d + 1 nc(x)�89 n 2 ,

and using the fact t ha t ,~, de(.), ~c(.) are bounded be tween posi t ive constants , an appl ica t ion of L e m m a 3 with l = 0 yields the following.

E

/ ( 1 - V(x, h))'~dh (~8) 0

< Ad+1 2 - 1 . . . 1

d--1 2 2 > =~+a/ (d + 1) +a-~A(B~-I) a-~ -r ( )a{-r d+~ 6q, ] n +a4r

- 1 , F(7~- 7) .(~c(z)~_~_r~ 1 a 1 ) , �9 " d - - 'ST- - 'S - -~ 2 - - O r 72

_ ( 1 + > 2(d+31 (d + 1) +a'~A(Bd-1) +a; -r "do(x)" n +

/~ d + l

where c~ > 0 is a sui table constant which, as well as the cons tant in O(.) depends on ~ and thus on A, but not on x.

414

By (56),

(29)

P. M. Gruber

,,c(~)

J (1 - V ( z , h ) ) " d h < we(u)(1 - V(z,e))" </37", E

where/3 > 0 depends only on C, while 7 depends on C, e and A and thus on ,k, and where 0 < 7 < 1.

From the representation of hE, in 3.1 and propositions (58) and (59) the following is obtained:

There are constants a, fl ,7 > 0,0 < 3' < 1, depending on C and A such that for n = 1 , 2 , . . . ,

a - z 2 2 t.A__

nd+-~ (he(u) - hE.(U)). (d+ 1)a-ZrA(Bd-') + ~ r . (dc(x)Z~ ' + ' ] r(&) oIx)] ]

- 1 �9 ( 1 + n ~ 7 '~

-> ~ld+3-----r a ) + -/~7" " /~ d + l 72

This holding for each/k, 1 < A < 2, implies that

2 nd-~[ ] = 1 + o(1) as n ~ c~, where o(-) may be chosen independent of X .

This concludes the proof of (55) and thus of part (ii) of Theorem 1.

3.4 P r o o f o f (iii)

3.4.1 Part (ii) yields our first proposition.

(60) Let C E C flC 2 with xo > 0. Then

= &S ~ ' - - - ~ O O .

The following counterpart of (60) will also be needed.

(61) Let P E C be a polytope. Then

1 H --, = o(~ (e. E.)) n~

a s n ---~ o o .

To see this, consider a vertex z of P and a supporting hyperplane Hc(u) of P such that Hc(u) f1 P = {x}. Then for all sufficiently small h > 0, say for 0 < h < ~, the cap Cp(u, h) has volume ah d for a suitable constant a > 0. Thus

ho(~,) - h~.(~,) = (1 - v (~ , h))"dh > 1 V(C)) 0 0

_ > ~ - for n = 1 ,2 , . . .


by 3.1 and Lemma 3, where fl > 0 is a suitable constant. This yields (61).

3.4.2 The convex bodies C 6 C M C 2 with tze > 0 and the polytopes P 6 C both form dense subsets of C. Thus (60), (61), the assumptions on the sequences (am) and (13=) in (iii), and Lemma 4 applied with T. , where

~.(C) = 6"(C, E,,),

together imply the assertion in (iii). The proof of part (iii) of Theorem 1 is complete.

4 O u t l i n e o f t h e p r o o f of T h e o r e m 2

The proof of Theorem 2 can be organized similarly to that of Theorem 1. Below we indicate the main steps of the proof.

4.1 A n expl ic i t r e p r e s e n t a t i o n of hEn. Let C E C and let dbac be a continuous density on bd C. By Od-1 and ad-: we denote the ( d - 1)-, resp. the (d - 2)-dimensional (surface area) measure. Then

~c(~}

h c ( u ) - h E , ( U ) = / ( 1 -A(u ,h ) ) ' ~dh f o r u 6 Sd-i 0

where

A(u, h) = / dbdc(y)dad-x(y). (bd C)nCc(u,h)

If dbdc = A(bd C) -1, then

m(bd C N Cc(u, h)) d(u, h) = A(bd C)

If C 6 C a and x E bd C, u = -no(x), then for A(u, h) we also write A(x, h).

4.2 P r o o f o f (i)

4.2.1 We first determine the area of a cap.

(62) There is a constant 6 > 0 such that the following hold. Let x E b d C . Then there are coefficients ed-a(x),..., ed+k-3(x) such that

d - - 1

A ( b d C M Co(x, h)) = ea_a(x)h-T +. . . + ed+k_3(x)h d§

+u(n = ) for O < h < 6 .

dd-a(-) > 0 and ed-a( ' ) , . . . ,ed+k-3( ' ) are continuous in x. The constant in O(.) may be chosen independent of x.

(25) holds a|so in the present case. Taylor's theorem together with the assumption in (i) that C 6 C k+l then shows the following.

416

(63)

P. M. Gruber

Let x 6 b d C , v 6 S d-2. Then for i = 1 , . . . , d - 1, there are coefficients bil(z, v ) , . . . , bik-l(x, v) such that

f , , (rv) = b, l(x ,v)r+ . . .+b,k_l(X,v)r k-1 +O(r k) for 0 < r < 0.

The coefficients are bounded, for fixed x they are continuous in v, and the constants in the O(.) symbols may be chosen independent of x, v.

Consider (26) and (27) and note that for fixed v the expression z = f ( r v ) is continuously differentiable in r, with positive derivative for r > 0. Hence the inverse function r is continuously differentiable in z with positive derivative for z > 0. Then by a result on asymptot ic expansions,

dr = -~c~(x,l v)z-~ + . . . + ~ - c k ( x , v ) z -~ + O(z ~~) for 0 < z < a.

For fixed x the coefficients c,(x, . ) , . . . , are continuous in v. This together with (63), (27) and the general binomial theorem or, instead of the lat ter , Taylor 's theorem, gives the next proposit ion.

(64) There is a constant 6 > 0 such that for all x 6 bd C and v 6 S d-2 there are coefficients ed-a(x, v) , . . . , ed+k-s(X, v) with

(1 + ~2 f , i .{rv) ) �89 = ed-3(x, v)z ~a +. . �9 + ed+k-5(x, V)Z d+~-~ i

+O(z dk~-~) for 0 < z < 5 ,

where z = f(rv).

The coefficients are bounded, for fixed x the}' are continuous in v, and the constant in O(.) may be chosen independent of x, v. For fixed x, the expression on the left hand side of our formula is continuous in v, z.

Now we prove the following.

(65) Let z 6 bd C. Then there are coefficients c~_l(x),. . . , ed+k-s(x) such tha t

d--1

A(bd C N Cc(x, h)) = ed_l(x)h-r + . . . + ed+k_3(x)h d~-3

+O(h 2 ) for O< h < 6.

The coefficients are bounded.

To see this, note that

(66) A(bdCnCc(z,h))= (l+~f.(u)).d~d-l(u) Cc(x,h)' i

/ ,d = (1 + ~ ] f , , (rv)2)~r -2drdad-2(v) rvECc(z,h) t i

uEsd--1

h dr

= / ( / + 0 S~-~ i


Considering (64), proposition (65) follows. An elementary argument (using inscribed paraboloids) shows that

(67) ed-l(') > 0.

Finally, similar reasoning as the one that led to (31) implies that

(68) de - i ( ' ) , . . . , ed+k-a(') are continuous.

Propositions (65), (67) and (68) together give (62).

4.2.2 The second step consists of expressing the height h of the cap Cc(x, h) in terms of its area.

(69) There is a constant r > 0 for which the following hold. Let x E bd C, u = -nc (x ) . Then there are coefficients gffx),.. . ,gk-l(x) such that for

= A(x, h) (= A(u, h) = A(bd C 0 Cc(x, h))) y_ n A(bdC)

we have y y 2 y k-1

h = h(x, ) = g2(X)(n)Z=f + . . . + gk_,(X)(n)Z:r

+ o ( ( ~ ) ~ ) for 0 < ~n -< r.

The coefficients are continuous, and the constant in 0(.) may be chosen independent of x. h is continuously differentiable as a function of } with positive derivative.

By (64) the integrand in the last integral in (66) is continuous in v, z. Hence A(bd CN Cc(x, h)) and thus ~ = A(x, h) is a continuously differentiable function of h. Since by (62) the derivative is positive for h > 0, the inverse function h is continuously differentiable in y for y > 0 and we may apply Lemma 2 to express h} in terms of y. Squaring this then gives (69).

4.2.3 The last part of the proof is close to that of the last part of (i) in Theorem 1.

4.3 P r o o f o f (ii) The proof is very similar to that of (ii) in Theorem 1. We indicate the only point where a slightly different argument is needed: (54) has to be replaced by the following proposition.

(70) Let C bd C. Then

A(bd C Q Cc(x, h)) ;2_ ~d+21 } 2~-~A(Be-1) h ~

~-r-~ ~c(~)~ for 0 < h < e

which follows from (53) and (47).

4.4 P r o o f o f (iii) Also in this case the proof is similar to that of (iii) in Theorem 1. (60) and (62) have to be replaced by (71) and (72), respectively:

418

(71) Let C E C N C 2 w i t h ~r > 0. T h e n

a s n ----~ (X).

P. M. Gmber

(72) Let P fi C be a po ly tope . T h e n

11 - o ( : ( c , E~ h a - 1

a s n ---+ ~ .

(71) is a consequence of pa r t (ii) of Theo rem 2 and (72) is ob ta ined in the same way as (62).

The out l ine of t he proof of Theo rem 2 is complete .

Acknowledgement For their many va luable hints I am obliged to Professors Chalk and Schnitzer , Dr .Ludwig and Dr .Re i t zne r .

References [1] Bs Vitale,R.A., Random convex hulls: Floating bodies and expectations, J. Ap-

prox. Th. 75 (1993) 130 - 135 [2] Blaschke,W., Vorlesungen fiber Differentialgeometrie, vol.II, 2nd ed., Springer, Berlin

1923 [3] Bruijn,N.G.de, Asymptotic methods in analysis, North-Holland, Amsterdam 1958, Dover,

New York 1981 [4] Buchta,Ch., Zufallige Polyeder - - Eine Ubersicht, in: E.Htawka, ed., Zahlentheoretische

Analysis, Lecture Notes in Math. 114, 1 - 13, Springer, Berlin 1985 [5] Gruber,P.M., Die meisten konvexen KSrper sind glatt, aber nicht zu glatt, Math. Ann.

229 (1977) 259 - 266 [6] Gruber,P.M., In most cases approximation is irregular, Rend. Sere. Mat. Univ. Politecn.

Torino 41 (1983) 19 - 33 [7] Gruber,P.M., Baire categories in convexity, in: P.M.Gruber, J.M.Wills, eds., Handbook

of convex geometry, vol.B, 1327 - 1346, North-Holland, Amsterdam 1993 [8] Gruber,P.M., Asymptotic estimates for best and stepwise approximation of convex bodies

I, II, llI, Forum Math. 5 (1993) 281 - 297, 521 - 538, to appear. Part IIl jointly with S.Glasauer

[9] Hurwitz,A., Courant,R., Funktionentheorie, 3rd ed., Springer, Berlin 1929 [10] Klee,V., Some new results on smoothness and rotundity in normed linear spaces, Math.

Ann. 139 (1959) 51 - 63 [1t] Leiehweiss,K., On inner parallel bodies in the equiafiine geometry, in: B.Fuchssteiner,

W.A.J.Luxemburg, eds., Analysis and geometry, 1 - 12, BI, Mannheim 1992 [12] Markushevieh,A.I., Theory of functions of a complex variable, vol.II, Prentice-Hall, En-

glewood Cliffs, N.J., 1965 [13] Miiller,J., On the mean width of random polytopes, Prob. Th. Rel. Fields 82 (1989) 33

- 37 [14] Schneider,R, Zur optimalen Approximation konvexer Hyperflgchen dutch Polyeder,

Math. Ann. 256 (1981) 289 - 301 [15] Schneider,R., Random approximation of convex sets, J.Microscopy 151 (1988) 211 - 227


[16] Schneider,R, Convez bodies: The Brunn-Minkowsky theory, Univ. Press, Cambridge 1993

[17] Schneider,R, Stochastic geometry, in: J.E.Goodman, J.O'Rourke, eds., Handbook of dis- crete and computational geometry, in preparation

[18] Schneider,R, Wieacker,J.A., Random polytopes in a convex body, Z. Wahrsch. Th. verw. Geb. 52 (1980) 69 - 73

[19] Simon,U., Zur Entwicklung der affinen Differentialgeometrie nach Blaschke, in: W.Blaschke, Gesammelte Werke, vol.4, 35 - 88, Thales Verlag, 1985

[20] Vinogradov,I.M. et al., eds., Encyclopedia of mathematics, Engl. Transl., Kluwer, Dor- drecht 1988- 1994

[21] Weil,W., Wieacker,J., Stochastic geometry, in: P.M.Gruber, J.M.Wills, eds., Handbook of convex geometry, vol.B, 1391 - 1438, North-Holland, Amsterdam 1993

[22] Whittaker,E.T., Watson,G.N., A course of modern analysis, 4th ed., Univ. Press, Cam- bridge 1927

expectation of random polytopes

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