Approximation of convex bodies by inscribed simplices of maximum volume

Abstract

The Banach-Mazur distance between an arbitrary convex body and a simplex in Euclidean n-space E n is at most n +2. We obtain this estimate as an immediate consequence of our theorem which says that for an arbitrary convex body C in E n and for any simplex S of maximum volume contained in C the homothetical copy of S with ratio n + 2 and center in the barycenter of S contains C. In general, this ratio cannot be improved, as it follows from the example of any double-cone.Denote by C n the family of all convex bodies and by M n the family of all centrally symmetric convex bodies in Euclidean n-dimensional space E n .Finding simplices of large volume in convex bodies has a long history. See, for instance, the papers by Blaschke (1917) on the maximum area of triangles in planar convex bodies, by MacKinney McKinney (1974) on simplices of maximum volume in centrally symmetric convex bodies, and the survey article Hudelson et al. (1996) by Hudelson, Klee and Larman on simplices of large volume in cubes.Let A be an (n − 1)-dimensional convex body and let I = ab be a perpendicular to the hyperplane containing A non-degenerated segment centered at a point of A. We call the convex hull conv(A ∪ I ) the double pyramid with base A and apices a and b. If B is a ball of E n − 1 in place of A and if the centers of B and I coincide, the double-pyramid is called a double-cone.

Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

“…Proof. In the proof of Proposition 5.6, we apply the main theorem of [21]: whenever one inscribes into a convex body in R d−1 a simplex of maximal volume, then scaling the body with ratio 1/(d + 1) from the barycenter of the simplex will make it contained in the simplex.…”

On the Beer Index of Convexity and Its Variants

“…For every d 2, there is a constant β = β(d) such that every convex set K ⊆ R d contains a simplex of measure at least βλ d (K) (see e.g. Lassak [13]). Therefore, Theorem 1.5 can be rephrased in the following equivalent form.…”

Minimum Convex Partitions and Maximum Empty Polytopes

“…From the other direction, García-López and Nicolás [20] proved that f 2 (n) ≥ 12n− 22 11 , for n ≥ 4, thereby improving an earlier lower bound f 2 (n) ≥ n + 2 by Aichholzer and Krasser [1]. Knauer and Spillner [29] have also obtained a 30 11 -factor approximation algorithm for computing a minimum convex partition for a given set S ⊂ R 2 , no three of which are collinear. There are also a few exact algorithms, including three fixed-parameter algorithms [17,21,38].…”

“…As far as the problem of Danzer and Rogers is concerned, one need not consider convex sets-it suffices to consider simplices-and for simplices the problems considered are much simpler. Specifically, every convex body C in R d , d ≥ 2, contains a simplex T of volume vol(T ) ≥ vol(C) /(d + 2) d [30]. That is, for fixed d, the largest empty simplex amidst n points in the unit cube [0, 1] d yields a constant-factor approximation of the largest volume convex body (polytope) amidst the same n points.…”