Umbrellas and Polytopal Approximation of the Euclidean Ball

Journal of Approximation Theory  AT3065 journal of approximation theory 90, 922 (1997) article no. AT963065

Umbrellas and Polytopal Approximation of the Euclidean Ball Yehoram Gordon* Department of Mathematics, Technion, Haifa 32000, Israel

Shlomo Reisner Department of Mathematics and School of Education-Oranim, University of Haifa, Israel; and Department of Mathematics and Computer Science, University of Denver, Denver, Colorado 80208

and Carsten Schutt  Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078; and Mathematisches Seminar, Christian-Albrechts Universitat, 24098 Kiel, Germany Communicated by D. Leviatan Received December 26, 1995; accepted in revised form May 31, 1996

There are two positive, absolute constants c 1 and c 2 so that the volume of the difference set of the d-dimensional Euclidean ball B d2 and an inscribed polytope with n vertices is larger than c 1 d vol d (B d2 ) n &2(d&1) for n(c 2 d ) (d&1)2.

 1997 Academic Press

We study here the approximation of a convex body in R d by a polytope with at most n vertices. There are many means to measure the approximation, the two most common are the Hausdorff distance or the symmetric * Yehoram Gordon was partially supported by the Fund for the Promotion of research at the Technion, and the France-Israel Arc en Ciel program. Shlomo Reisner was supported by the France-Israel Arc en Ciel program and NSF Grant DMS-9626749.  Carsten Schutt has been supported by NSF Grant DMS-9301506.

9 0021-904597 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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10

GORDON, REISNER, AND SCHUTT

difference metric. The Hausdorff distance between two convex bodies K and C is d H (K, C)=max[max min &x& y&, max min &x& y&] x#C y#K

y#K x#C

where &x& is the Euclidean norm of x. The symmetric difference metric is the volume of the difference set. d S (K, C)=vol d (K q C). Bronshtein and Ivanov [1] and Dudley [1, 2] showed that for every convex body K in R d there is a constant c=c(K, d ) such that for every n there is a polytope P n with at most n vertices and d H (K, P n )cn &2(d&1). This can be used to show the same estimate for the symmetric difference metric. Gruber and Kenderov [9] showed that the inverse inequality holds if K has a C 2-boundary: d S (K, P n )cn &2(d&1). Macbeath [10] showed that the approximation of a convex body is always better than that of the Euclidean ball. Gruber [8] obtained an asymptotic formula. If a convex body K in R d has a C 2-boundary with everywhere positive curvature, then we have inf [d S (K, P n ) | P n /K and P n has at most n vertices] 1 t del d&1 2

|

}(x) 1(d+1) d+(x)

K

1 n

\+

2(d&1)

where del d&1 is a constant that is connected with triangulations. In [5] and [6], it was shown constructively that for all dimensions d, all convex bodies K, and all n2 there is a polytope P n with n vertices that is contained in K such that vol d (K)&vol d (P n )c 1 d vol d (K) n &2(d&1) where c 1 is a numerical constant. This estimate can also be derived from [1] and [2, 3]. So the question was whether the factor d was necessary, or, in other words, what is the order of magnitude of the constant del d . The result in this paper shows that there are absolute positive constants c 1 and c 2 with c 1 del d c 2 .

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UMBRELLAS AND THE EUCLIDEAN BALL

11

In fact, we have del d&1 

32 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

this follows from estimate (1) below. In this paper we want to show that there is a universal constant c>0 such that the volume of the difference set of the d-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than c d vol d (B d2 ) n &2(d&1). We want to reduce the computation of the volume of the difference set to that of the following set: The set between a d&1-dimensional face of the polytope and the boundary of the sphere. Intuitively it is clear that the faces should be simplices and that the polytope should have rather regular features. This leads us to the assumption that the volume of the set between a d&1-dimensional face of the polytope and the boundary of the sphere equals in average approximately the surface area of the face times the height of the cap of the Euclidean ball that is determined by that face. There are two technical difficulties. The number of faces does not necessarily correspond to the number of vertices. In fact, a heuristic argument shows that the number of faces is of the order of the number of vertices times d d2. Secondly, although we may assume that the faces are simplices, we may not assume that they are regular or close to regular. This is expressed in the following way. If F is a face and H the hyperplane containing F then the distance of the centers of gravity of F and H & B d2 may be large. Hyperplanes are usually denoted by H and the closed halfspaces associated with H by H + and H &. H(x, !) is the hyperplane that passes through x and is orthogonal to !. The d&1-dimensional faces of a polytope in R d are denoted by F j . The denotes the halfspace hyperplanes containing F j are denoted by H j . H + j containing P. For a polytope P that is contained in B d2 the height or width of B d2 & H & j is h j and the radius of B d2 & H j is r j . cg(M) is the center of gravity of the set M. [A, B] denotes the convex hull of the sets A and B. The radial projection rp(M) of a set M in B d2 is rp(M)=[! # B d2 | [0, !] & M{<].

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12

GORDON, REISNER, AND SCHUTT

Theorem 1. There are two positive constants c 1 and c 2 so that we have for all d, d2, and all n, n(c 2 d ) (d&1)2, and all polytopes P n that are contained in the Euclidean unit ball B d2 and have n vertices vol d (B d2 )&vol d (P n )c 1 d vol d (B d2 ) n &2(d&1). In particular we have by Theorem 1 that vol d (B d2 )&vol d (P n )

c1 vol d (B d2 ) c2

if n(c 2 d ) (d&1)2. Lemma 2.

(i)

For all x, 0
1(x+1)=- 2? x x+12 exp vol d (B d2 )=

(ii)

%

\ &x+12x+ .

? (d&1)2(2e) d2 ? d2  . 1((d2)+1) d (d+1)2

The following lemma is due to Bronshtein and Ivanov [1] and Dudley [2, 3]. Lemma 3. For all dimensions d, d2, and all natural numbers n, n2d, there is a polytope Q n that has n vertices and is contained in the Euclidean ball B d2 such that d H (Q n , B d2 )

16 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1).

In particular, since a Q n which satisfies the hypothesis of Lemma 3 contains the Euclidean ball of radius 1&d H (Q n , B d2 ), it follows that d S (Q n , B d2 )vol d (B d2 )[1&(1&d H (Q n , B d2 )) d ]

{ \

vol d (B d2 ) 1& 1&

16 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

d

2(d&1)

n &2(d&1)

+= (1)

and

{

1&

16 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1)

=

d&1

vol d&1(B d2 )vol d&1(Q n ). (2)

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UMBRELLAS AND THE EUCLIDEAN BALL

13

We have that vol d&1(B d2 )=d vol d (B d2 )=d =d - ?

? d2 1((d2)+1)

1((d&1)2+1) vol d&1(B d&1 ) 2 1((d2)+1)

d - ? vol d&1(B d&1 ). 2 Since d 2(d&1) 4 and (1&t) d 1&dt we get from (1) &2(d&1) d d S (Q n , B d2 )(1&(1& 64 ) ) vol d (B d2 ) 7 ?n &2(d&1) vol d (B d2 ).  64 7 ? dn

(3)

(d&1)2 Similarly we get from (2) that we have for n( 128 . 7 ? d)

vol d&1(B d2 )2 vol d&1(Q n ).

(4)

For the sake of completeness we include the proof of Lemma 3. The arguments are from [1]. Proof. For every n there is a % n >0 and a set [x 1 , ..., x n ]/B d2 so that for all i{ j we have &x i &x j &% n and so that for every x # B d2 there is i such that &x&x i &% n . We choose Q n to be the convex hull of [x 1 , ..., x n ]. We have d H (Q n , B d2 ) 12 % 2n . If not, then there is x # B d2 such that the Euclidean ball with radius 12 % 2n and center x and Q n have an empty intersection. By the theorem of HahnBanach there is a hyperplane separating Q n and B d2(x, 12 % 2n ). This hyperplane cuts off a cap of height greater than 12 % 2n . The point at the top of this cap has a distance greater than % n from all x i , i=1, ..., n. This cannot be. Now we estimate % n from above. The caps B d2 & H & ((1& 18 % 2n ) x i , x i )

i=1, ..., n

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14

GORDON, REISNER, AND SCHUTT

have disjoint interiors. Therefore we get n

vol d&1(B d2 ) : vol d&1 (B d2 & H &((1& 18 % 2n ) x i , x i )) i=1 1 2 d&1 n( 12 % n - 1& 16 % n) vol d&1(B d&1 ). 2

We obtain 1 %n 2



1&

1 2 1 vol d&1(B d2 ) % n 16 n vol d&1(B d&1 ) 2

\

+

1(d&1)

For n=2d we get that % n - 2. Indeed, just consider the set [e 1 , ..., e d , &e 1 , ..., &e d ]. Thus it follows %n 2

7 1 vol d&1(B d2 )  8 n vol d&1(B d&1 ) 2

 \

+

1(d&1)

and thus 1 2 16 1 vol d&1(B d2 ) %  2 n 7 n vol d&1(B d&1 ) 2

\

Lemma 4.

d

R+

(ii)

|

\

d

R+

(iii)

. K

For k=0, 1, 2, .. . and d=1, 2, . . . we have

(i)

|

+

2(d&1)

\

d

: yi i=1

d

+

k

\ \

d

2

exp & : y i

+ \ \

i=1

d

2

: y 2i exp & : y i i=1

+ + dy=

i=1

++

dy=

1((k+1)2) . 2(d&1)!

d2 d 1 . 2(d+1)! 2

\+

For i{ j we have

|

d R+

\ \

d

y i y j exp & : y i i=1

2

++

dy=

1(d2) . 4(d+1)(d&1)!

Proof. This is an easy consequence of Fubini theorem with the change of variables x d = di=1 y i and x i = y i , i=1, ..., d&1. K For the following lemma compare also [12]. Lemma 5. Let x 1 , ..., x d be points on the Euclidean sphere of radius 1, S the simplex [x 1 , ..., x d ], and rp(S) the radial projection of S, i.e., the

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15

UMBRELLAS AND THE EUCLIDEAN BALL

spherical simplex of the points x 1 , ..., x d . Let X be the matrix whose columns are the vectors x 1 , ..., x d . Then we have vol d&1(rp(S))=

2 |det(X)| 1(d2)

|

exp( & y tX tXy) dy

d

R+

and vol d ([0, rp(S)])= Proof.

1 |det(X)| 1((d2)+1)

|

d

exp( & y tX tXy) dy.

R+

We have vol d (B d2 )=

? d2 1((d2)+1)

and

|

Rd

2

e &&z& dz=? d2.

Therefore we get vol d ([0, rp(S)])=

1 1((d2)+1)

|

2

e &&z& dz. [z=t! | ! # S and t # R + ]

Using the substitution z=Xy we get the latter expression equals 1 |det(X)| 1((d2)+1)

|

t

d

t

e & y X Xy dy.

K

R+

Lemma 6. Let x 1 , ..., x d be points on the Euclidean sphere of radius 1, S the simplex [x 1 , ..., x d ], and let rp(S) be the radial projection of the simplex S. Let H be the hyperplane containing the simplex [x 1 , ..., x d ] and r the radius of the d&1-dimensional Euclidean ball H & B d2 . Then we have vol d ([0, rp(S)])&vol d ([0, S])

d2 1 d : xi 1& 2(d+1) d i=1

\ "

2

" + vol ([0, S]) d

and vol d ([0, rp(S)])&vol d ([0, S])

d - 1&r 2 1 d : xi 1& 2(d+1) d i=1

\ "

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2

" + vol

(S).

d&1

16

GORDON, REISNER, AND SCHUTT

Proof.

By Lemma 5 we have

vol d ([0, rp(S)])&vol d ([0, S]) =

1 |det(X)| 1((d2)+1)

|

d R+

exp( & y tX tXy) dy&

|det(X)| . d!

By Lemma 4(i) with k=0 the last expression equals 1 |det(X)| 1((d2)+1) =

|

d

R+

{

\ \

d

2

exp( & y tX tXy)&exp & : y i i=1

+ += dy

1 |det(X)| 1((d2)+1)

|

_

d

R+

{ \\ exp

d

: yi i=1

+

2

+ = \ \

d

2

& y tX tXy &1 exp & : y i i=1

+ + dy.

We use now the inequality 1+te t and get that the above expression is greater than or equal to 1 |det(X)| 1((d2)+1) =

|

d

R+

{\

d

: yi i=1

+

2

= \ \

d

& y tX tXy exp & : y i

d 1 |det(X)| : (1&( x i , x j ) ) 1((d2)+1) i, j=1

|

i=1

2

+ + dy

y i y j exp( &( y ty) 2 ) dy.

d

R+

Since we have ( x i , x i ) =1 for i=1, ..., d, we get by Lemma 4(iii) for the above expression =

d 1 1(d2) |det(X)| : (1&( x i , x j ) ) 1((d2)+1) 4(d+1)(d&1)! i, j=1 d 1 |det(X)| d 2 & : x i 2(d+1)! i=1

2

\ " "+ d - 1&r 1 = 1& " d : x " + vol (S). 2(d+1) \

=

2

d

2

i

d&1

K

i=1

Lemma 7. Let A be a measurable subset of B d2 such that the center of gravity of A is contained in a cap of height 2, 21. Then there is a cap C of height 22 so that 2 vol d (C & A)vol d (A).

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UMBRELLAS AND THE EUCLIDEAN BALL

17

Proof. Let us consider B d2 with the origin situated at the ``south pole'', 2 2 i.e. B d2 =[x=(! 1 , ..., ! d ) |  d&1 i=1 ! i +(! d &1) 1]. W.l.g. assume that the center of gravity of the set A is on the ! d axis at the point (0, ..., 0, 2) where 0<21. Let a(t) be the d&1-dimensional Lebesgue measure of the intersection of A with the hyperplane [x | ! d =t]. Then vol d (A)= 20 a(t) dt and 2=(1vol d (A))  20 ta(t) dt. Let C be the cap B d2 & [x | ! d 22]. We obtain 2vol d (C & A)=2

|

a(t) dt

t<22

=2vol d (A)&2

|

a(t) dt

t22

2vol d (A)&

t a(t) dt& t22 2

|

t a(t) dt t<22 2

|

=vol d (A). K Lemma 8. Let P n be a simplicial polytope with vertices x 1 , ..., x n that are elements of B d2 . Let F j , j=1, ..., m be the d&1-dimensional faces of P n , H j the hyperplane containing F j , h j the height of the cap B d2 & H & j , and r j the radius of B d2 & H j . Let N be the set of integers j so that hj 

1 vol d&1(P n ) 1 8 vol d&1(B d2 ) 4n

\

+

2(d&1)

.

Then we have vol d&1 Proof.

\

1 . F j  vol d&1(P n ). 4 j#N

+

We put Ni =[ j # N | x i # F j ]

i=1, ..., n

and \=

1 vol d&1(P n ) 1 8 vol d&1(B d2 ) 4n

\

+

2(d&1)

.

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18

GORDON, REISNER, AND SCHUTT

Since h j \ we have that  j # Ni F j is contained in B d2(x i , 2 - 2\).  j # Ni F j is a subset of the boundary of the convex set P n & B d2(x i , 2 - 2\). Thus we get vol d&1

\ . F + vol j

d&1

((P n & B d2(x i , 2 - 2\))).

j # Ni

Since P n & B d2(x i , 2 - 2\) is a convex subset of the convex set B d2(x i , 2 - 2\) we get vol d&1

\ . F + (8\)

(d&1)2

j

vol d&1(B d2 )

j # Ni

1 vol d&1(P n ). 4n

Therefore we get vol d&1

\

+

\

n

+ 1 \ . F + 4 vol

. F j =vol d&1 . j#N

. Fj

i=1 j # Ni

n

 : vol d&1

j

d&1

(P n ).

K

j # Ni

i=1

Lemma 9. Let P n be a simplicial polytope with vertices x 1 , ..., x n that are elements of B d2 . Let F j , j=1, ..., m be the d&1-dimensional faces of P n , H j the hyperplane containing F j , h j the height of the cap B d2 & H & j , and r j the radius of B d2 & H j . Assume that we have for all j, j=1, ..., m hj 

16 vol d&1(B d2 ) 2 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1)

and assume that vol d&1(B d2 )2 vol d&1(P n ). Let M be the set of integers j so that &cg(F j )&cg(H j & B d2 )&

2 22 &1 rj . 2 22

Then we have vol d&1

\

1 . F j  vol d&1(P n ). 4 j#M

+

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19

UMBRELLAS AND THE EUCLIDEAN BALL

Proof.

We put %= 

vol d&1(B d2 ) 16 2 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1)

16 (2d - ?) 2(d&1) n &2(d&1). 7

Since k j % we have for all j, j=1, ..., m r j - 2%. We have that cg(F j ) is contained in a cap of height 2 &22r j of the d&1dimensional Euclidean ball H j & B d2 . By Lemma 7 there is a subset F j of F j so that F j is contained in a cap of height 2 &21r j and vol d&1(F j )2 vol d&1(F j ). Thus the diameter of F j is less than 2 &9r j - 2%512. The set of all integers j such that x i # F j is denoted by Mi . We have that  j # Mi F j is a subset of the boundary of the convex set P n & B d2(x i , 2 &9 - 2%) and has a smaller surface area than B d2(x i , 2 &9 - 2%). vol d&1

\

j # Mi

- 2% 512

d&1

+ \ + vol (B ) 4d - ? - 32  vol n \ 512 - 7 +

. F j 

d 2

d&1

d&1

d&1

(P n ).

Since d2 d&1 we get that the latter expression is smaller than 4 -? -2 n 128

\ +

d&1

vol d&1(P n )

- 2? 1 vol d&1(P n ) vol d&1(P n ). 32n 8n

Therefore we get vol d&1

\

+

. F n =vol d&1 j#M

\

n

.

i=1 j # Mi

n

2 : vol d&1 i=1

n

+

. F j  : vol d&1

\

i=1

\ . F+ j

j # Mi

1 . F j  vol d&1(P n ). 4 j # Mi

+

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20

GORDON, REISNER, AND SCHUTT

Proof of Theorem 1. We consider numbers of vertices n such that (d&1)2 . Let P n be a polytope with n vertices so that vol d (B d2 )& n( 512 7 ?d ) vol d (P n ) is minimal. Let Q n be a polytope with n vertices so that d H (B d2 , Q n ) is minimal. By Lemma 3 we have that for all j d H (B d2 , Q n )

16 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1).

We consider now the convex hull of P n and Q n . P=[P n , Q n ]. P has at most 2n vertices. Its d&1-dimensional faces are denoted by F j , j=1, ..., m. H j is the hyperplane containing F j , h j the height of the cap d B d2 & H & j , and r j the radius of B 2 & H j . We may assume that P is simplicial. We have that h j d H (B d2 , Q n )

16 vol d&1(B d2 ) 7 vol d&1(B d&1 ) 2

\

+

2(d&1)

n &2(d&1).

By the assumption on n we have that h j  18

and

r j =- 2h j &h 2j  12 .

(5)

Also we have by (4) that vol d&1(B d2 )2 vol d&1(Q n )2 vol d&1(P). We apply Lemmas 8 and 9 to P that has at most 2n vertices. Thus a factor 2 enters the estimates. Let L be the set of integers j so that 1 vol d&1(P n ) 1 8 vol d&1(B d2 ) 8n

\

+

2(d&1)

h j 

16 vol d&1(B d2 ) 1 7 vol &1(B d&1 )n 2

\

+

2(d&1)

(6)

and &cg(F j )&cg(H j & B d2 )&<

2 22 &1 rj . 2 22

(7)

We have vol d&1

\

1 . F j  vol d&1(P). 2 j#L

+

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(8)

21

UMBRELLAS AND THE EUCLIDEAN BALL

We apply Lemma 6 vol d (B d2 )&vol d (P n )vol d (B d2 )&vol d (P)  : (vol d ([0, rp(F j )])&vol d ([0, F j ])) j#L

- 1&r 2j (1&&cg(F j )& 2 ) vol d&1(F j ). 4 j#L

:

By (5) we have r j  12 and get that the latter expression is greater than 1 (1&&cg(F j )& 2 ) vol d&1(F j ). 8 j#L :

We have &cg(F j )& 2 =(1&h j ) 2 +&cg(F j )&cg(H j & B d2 )& 2. By (7) we get for j # L 2 22 &1 rj 2 22

2

\ + 2 &1 =1&(1&h ) & \ 2 + (2h &h )

1&&cg(F j )& 2 1&(1&h j ) 2 &

22

2

2

j

22

j

2 j

=(2 &21 &2 &44 )(2h j &h 2j )2 &21h j . Therefore vol d (B d2 )&vol(P)

1 : h j vol d&1(F j ). 2 24 j # L

By (6) we get that this expression is greater than 1 vol d&1(P) 1 2 27 vol d&1(B d2 ) 8n

\

+

2(d&1)

: vol d&1(F j ), j#L

and by (8) this expression is greater than 1 vol d&1(P) 1 2 29 vol d&1(B d2 ) 8n

\



+

2(d&1)

vol d&1(P)

1 1 vol d&1(B d2 ) n &2(d&1) = 36 d vol d (B d2 ) n &2(d&1). 2 36 2

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GORDON, REISNER, AND SCHUTT

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