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ON THE VOLUME OF THE CONVEX HULL OF TWO CONVEX BODIES

G. Horváth, Ákos, Zsolt Lángi

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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. 1.

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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. 1.

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Available 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. 1.

Key concepts: Convex hull, Convex body, Mixed volume, Mathematics, Hyperplane, Combinatorics, Convex set, Convex polytope

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