Polyhedral approximation of smooth convex bodies

JOURNAL

OF MATHEMATICAL

Polyhedral

ANALYSIS

AND

APPLICATIONS

Approximation

128,

47@474

of Smooth

(1987)

Convex

Bodies

ROLF SCHNEIDER Mathematisches Institut, Urziversitiit Freiburg i. Br., Alberistrassr 23 b, D-7800 Freiburg i. Br., West Germany Submitted by Ky Fan Received February

19, 1986

In [3], McClure and Vitale have obtained a series of precise asymptotic results on the approximation of smooth convex curves in the plane by inscribed or circumscribed polygons with a given number of vertices. In the present note we generalize one of their results to higher dimensions. Let S” be the set of convex bodies (compact convex subsets with interior points) in d-dimensional Euclidean space W’ with scalar product ( ., ) and norm /I I I/. For C E X” we denote by 9’;,,,(C) the set of (convex) polytopes containing C with at most n facets. Let 6(C, Y&,)=inf{G(C,

P): PE~);~)(C)),

where 6(K, L) = max { sup inf 11x - y 11,sup inf IIx - y I() CEK

L.E~

rtl

>c=K

defines the Hausdorff metric on 2’“. By Xf we denote the set of all convex bodies C whose boundary aC is a differentiable hypersurface with regular, three times continuously everywhere positive curvatures. For C E .Tg, define

where K is the Gauss-Kronecker curvature (product of the curvatures) of X and dA denotes the surface area element. ICY= #‘*/I’( (d/2) + 1) is the volume of the d-dimensional unit ball the density of a least dense covering of Rd by congruent balls Rogers [4, pp. 16-201). THEOREM.

Zf C E Xf,

fhen

lim n2/cd- ‘)d(C, &PP;J= $(C)*lcdp “. n-m 470

principal Further, and od is (see, e.g.,

POLYHEDRAL

APPROXIMATION

471

For d = 2, this reduces to case (i) in Theorem 1 of McClure and Vitale [3] (see also Fejes Toth [l, footnote 21). These authors obtained a number of analogous results, also for inscribed polygons and for other measures of deviation; their smoothness assumptions are slightly weaker. On the other hand, their methods are restricted to the two-dimensional case, and for most of their results no higher-dimensional versions are known (see Gruber [2] for a useful survey over approximation problems for convex bodies). An analog of the theorem above for approximation by polytopes contained in C with a given number of vertices was proved by Schneider [S]. If C is a ball, either result is easily deduced from the other by means of polarity, but for general C formula (1) requires a proof of its own. This proceeds by a variant of the method applied in [S], and it is dual to the latter proof only in a vague sense. For the proof of (1) we need some estimates and arguments from [S]. Let C E X$ be given. As in [S], we introduce on dC the Riemannian metric given by the second fundamental form of differential geometry. The corresponding geodesic distance of the points x, y E aC is denoted by y(x, y). The set B(p, Y)= {xEX: y(x,p) 0. Let p E X be given. By T, we denote the tangent hyperplane to X at p and by ePthe interior unit normal vector of C at p. The distance of a point x E X from Tp is denoted by h(x, p). For h>O, let F(p,h)= { xEaC: h(x,p) 0 such that, for all p E dC and all h with 0
(2) (3)

Proof: For p E X, we denote by D, the relative interior of the image of C under orthogonal projection into T,. There is a function zp of class C3 on D, such that x + P(x) ePE K for x E D, and the open segment between x and x + z”(x) ePdoes not meet C. For given 1 with 0 < I < 1 choose h, as in the proof of Hilfssatz 1 in [S]. Let 0 < h < hl and p E X be given. As in [S], the point p is chosen as the origin of R’, and partial derivatives of zp refer to some fixed Cartesian coordinate system in T,. Let x E B(p, A(2h)“‘), hence y(x, p) < 1(2h)“‘. If h(x, p) > h,, then there exists a point y E aC (on a shortest curve in aC joining p and x) satisfying Y(Y, P) < Y(.-GPI and h(y, p) = h,. From 15, (6)l it follows that y(y,p) 2

472

ROLF SCHNEIDER

A(2h(y, p))“’ = A(2h,)“’ > y(x, p), a contradiction. Hence h(x, p) 0 there exists rj > 0, independent of p, such that P’(x,)
x&x&

which yields zp(xo) = h + 1 [f~;~(O~x~) - tzk(O,x,)] I, i. k

x;x&.

Observing that h(x, + zP(xO)ep, p) < h,, we deduce from the proof of Hilfssatz 1 in [S] that I zp(xo) -h I 6 0, IIxo II3

(4)

with a constant a, independent of p. From [IS, (13) and (IS)] it follows that IIx0 II2 d vP(xo)

(5)

with a contant a2 independent of p. We can obviously choose 0
Since x0 + zP(xo) eP was an arbitrary boundary point of S(p, h), this shows that

0,

Ah)= VP, h) = F(p, A- ‘h).

(6)

POLYHEDRAL APPROXIMATION

413

Finally, we choose q, = min { gn, Ah,}. With this choice, the inclusions of (6) and (2) imply those of (3). This proves Lemma 1. For p > 0, define k(C, p) as the smallest number k such that 8C can be covered by k geodesic balls B(p;, p) of radius p. The following lemma was proved in [S] . LEMMA

2. lim,,,

k(C, p) pd- ’ = a(C).

For E> 0, define &(C, E) as the smallest number n for which there exists a polytope P E P;n,( C) satisfying 6(C, P) < E. LEMMA 3.

For 0 <

E <

qr,

k(C, ~-3’2(2~)“2)0f’(c,

c)
A3’*(2q1’*).

(7)

Proof: There is a polytope P containing C with m = mC(C, E) facets F, , .... F,,, satisfying 6(C, P) < E. Without loss of generality we may assume that P is circumscribed to C; let pi be the point where F, touches C. Let x E dC. Choose y on the exterior normal ray to 8C at x such that 11y-x 11= E. Suppose that S(x, E) contains none of the points p,, .... pm. Then for each in { 1, .... m}, the affne hull of the facet F, does not (weakly) separate y and C, hence y is an interior point of P. Therefore, P contains a point at distance greater than E from C, a contradiction. Hence, S(x, E) contains some pj. From (3) we have y(p,, x)
f) ff+(C,Pi), 1=-I

where H+ (C, pi) is the closed supporting halfspace (containing C) to C at pi. Then P is a polytope circumscribed about C with at most k facets (it is clear that P is bounded if E is sufficiently small). Suppose that 6( C, P) > E. Then there is a vertex u of P at distance E’ > E from C. Let x be the point in C nearest to u. Then S(x, E) contains no pi, since otherwise o $ P. By (3), B(x, A3’*(2s)‘/*) contains no pi, hence x $ B(p,, 123/2(2~)1’2)for i = 1, .... k, a contradiction. It follows that 6(C, P) GE, hence we have proved the righthand inequality of (7) and thus Lemma 3.

474

ROLF SCHNEIDER

Precisely as in [S] one now deduces from Lemmas 2 and 3 that lim m’( C, E)(~E)+ ‘)‘* = a(C),

E -0

and this is obviously equivalent to (1).

REFERENCES 1. L. FEJES T&H, Approximation by polygons and polyhedra, Bull. Amer. Math. Sot. 54 (1948), 431438. 2. P. M. GRUBER, Approximation of convex bodies, in “Convexity and Its Applications” (P. M. Gruber and J. M. Wills, Eds.), pp. 131-162, Birkhauser, Basel/Boston/Stuttgart, 1983. 3. D. E. MCCLURE AND R. A. VITALE, Polygonal approximation of plane convex bodies, J. Math. Anal. Appl. 51 (1975), 326358. 4. C. A. ROGERS, “Packing and Covering,” University Press, Cambridge, England, 1964. 5. R. SCHNEIDER, Zur optimalen Approximation konvexer Hyperfkichen durch Polyeder, Math. Ann. 256 (1981), 289-301.