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Linearity of the volume. Looking for a characterization of sausages

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Date
2014-07-30
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Authors
Saorín Gómez, Eugenia ; Yepes Nicolás, Jesús
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Facultad de Matemáticas
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Publisher
Elsevier
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Matemáticas
DOI
https://doi.org/10.1016/j.jmaa.2014.07.060
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info:eu-repo/semantics/article
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Abstract
Let K,E,L be convex bodies, dim⁡L≤1 and K=L+E, a sausage. In this case vol(λK+(1−λ)E)=λvol(K)+(1−λ)vol(E). We prove that under the sole assumption that K and E have an equal volume projection (or a common maximal volume section), if the above equality holds for just one value in (0,1), then K=L+E with dim⁡L≤1. However, even having equality for all λ∈[0,1], if no extra assumption on K,E is done, such a characterization is not possible. This problem is connected with a conjecture relating the roots of the Steiner polynomial of a pair of convex bodies to their relative inradius. Counterexamples for the general case are explicitly given. In the same line, a counterexample to a conjecture by Matheron on inner parallel bodies is also shown.
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Linearity of volume , Sausages , Brunn–Minkowski inequality , Roots of Steiner polynomial , Inner parallel body , Linearity of determinant
Citation
Journal of Mathematical Analysis and Applications, Volume 421, Issue 2, 2015, Pages 1081-1100, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2014.07.060.
URI
http://hdl.handle.net/10201/212441
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