Closed convex sets with an open or closed Gauss range
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Closed convex sets with an open or closed Gauss range Juan Enrique Martínez-Legaz1,2
· Cornel Pintea3
Received: 8 October 2019 / Accepted: 1 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020
Abstract We characterize the closed convex subsets of Rn which have open or closed Gauss ranges. Some special attention is paid to epigraphs of lower semicontinuous convex functions. Keywords Closed convex set · Gauss map · Gauss range · Motzkin decomposable convex set · Minkowski convex set Mathematics Subject Classification 52A20 · 53A07
1 Introduction The Gauss map of a closed convex set C ⊆ Rn , as defined by Laetsch [11] (see also [1]), generalizes the S n−1 -valued Gauss map of an orientable regular hypersurface of Rn . While the shape of such a regular hypersurface is well encoded by the Gauss map, the range of this map, equally called the spherical image of the hypersurface, is used to study various aspects of the smooth convex hypersurfaces both in the finite dimensional setting [6,16,17] and in the infinite dimensional setting [3–5,11] as well. The study of
Dedicated to Prof. Marco A. López on the occasion of his 70th birthday. Juan Enrique Martínez-Legaz acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Program for Centers of Excellence in R&D (CEX2019-000915-S). He is affiliated with MOVE (Markets, Organizations and Votes in Economics). Cornel Pintea was supported by a Grant of the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0190, within PNCDI III.
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Juan Enrique Martínez-Legaz [email protected]
1
Department of Economics and Economic History, Universitat Autònoma de Barcelona, Bellaterra, Spain
2
BGSMath, Barcelona, Spain
3
Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Cluj-Napoca, Romania
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J. E. Martínez-Legaz, C. Pintea
closed convex sets through their Gauss ranges encounters an interior versus boundary dichotomy for the points of the Gauss range, which is reflected in a bounded versus unbounded dichotomy at the facial structure level. Indeed, every interior point of the Gauss range produces an exposed bounded proper face and every intersection point of the Gauss range with its boundary produces an unbounded proper exposed face of the involved closed convex set, whenever this closed convex set has nonempty interior. On the other hand it is self-evident that the lack of border points in the Gauss range, which happens when the Gauss range is open, entails the lack of exposed unbounded proper faces. These two situations, i.e. open versus closed for the Gauss ranges, are extreme for every size evaluation tool of the intersection between the Gauss range and its boundary. This intersection set is expected to encode the amount of unbounded exposed proper faces. At one extreme we have those unbounded closed convex sets with several
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