On the volume of the convex hull of two convex bodies

Abstract

Abstract. In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean n-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.

Volume of Convex Hull of Two Bodies and Related Problems

“…G.Horváth and Zs. Lángi in [19] introduced the following quantity. Furthermore, if S is a set of isometries of R d , we set…”

“…The examination of the volume of the convex hull of two congruent copies of a convex body in Euclidean d-space (for special subgroups) investigated systematically first by Rogers, Shepard and Machbeth in 1950s (see in [40], [41] and [34]). Fifty years later a problem similar to that of the simplices arose that lead to new investigations by new methods which obtained fresh results (see in [19], [18], [21]). In particular, a related conjecture of Rogers and Shephard has been proved in [19].…”

On a normed version of a Rogers–Shephard type problem

“…is the length of a longest chord of K in the direction of u, and w K (u ⊥ ) is the width of K in the direction perpendicular to u (cf. also the proof of Theorem 1 in [9]). Observe that for any direction u, we have d K (u) = d M (u) and w K (u) = w M (u), which yields that minimizing c Bus tr (K), over the class of convex disks with a given central symmetral, is equivalent to minimizing λ(K) within this class.…”

Bounds for Totally Separable Translative Packings in the Plane

“…is the length of a longest chord of K in the direction of v, and w K (v ⊥ ) is the width of K in the direction perpendicular to v. The observation that this property is equivalent to the fact that bd K is a Radon curve can be found, for example, in the proof of Theorem 2 of [11]. Now consider the case that Q is o-symmetric, but K is not necessarily.…”