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Colorful theorems for strong convexity

Authors: Andreas F. Holmsen and Roman Karasev
Journal: Proc. Amer. Math. Soc. 145 (2017), 2713-2726
MSC (2010): Primary 52A35, 52A20
Published electronically: November 30, 2016
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Abstract: We prove two colorful Carathéodory theorems for strongly convex hulls, generalizing the colorful Carathéodory theorem for ordinary convexity by Imre Bárány, the non-colorful Carathéodory theorem for strongly convex hulls by the second author, and the ``very colorful theorems'' by the first author and others. We also investigate if the assumption of a ``generating convex set'' is really needed in such results and try to give a topological criterion for one convex body to be a Minkowski summand of another.

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Andreas F. Holmsen
Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Daejeon 305-701, South Korea

Roman Karasev
Affiliation: Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700 – and – Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994

Keywords: Carath\'eodory's theorem, Helly's theorem, strong convexity
Received by editor(s): September 29, 2015
Received by editor(s) in revised form: July 13, 2016, and July 31, 2016
Published electronically: November 30, 2016
Additional Notes: The first author was supported by Swiss National Science Foundation Grants 200020-144531 and 200021-137574
The second author was supported by the Russian Foundation for Basic Research grants 15-31-20403 (mol_a_ved) and 15-01-99563 (A) and by ERC Advanced Research Grant No. 267195 (DISCONV)
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

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