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.
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Acknowledgments
The support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences is gratefully acknowledged. The authors are indebted to Endre Makai, Jr. for pointing out an error in an earlier version of the proof of Theorem 1, and to an unknown referee for many helpful remarks.
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Communicated by A. Constantin.
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G. Horváth, Á., Lángi, Z. On the volume of the convex hull of two convex bodies. Monatsh Math 174, 219–229 (2014). https://doi.org/10.1007/s00605-013-0526-x
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DOI: https://doi.org/10.1007/s00605-013-0526-x