Journal of Convex Analysis 30 (2023), No. 4, 1203--1216
Copyright Heldermann Verlag 2023
On the Gauss Range of a Closed Convex Set
Juan Enrique Martínez-Legaz
Dep. d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, Barcelona Graduate School of Mathematics, Spain
JuanEnrique.Martinez.Legaz@uab.es
Cornel Pintea
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania, Romania
cpintea@math.ubbcluj.ro
The aim of this paper is twofold. One one hand we study how the Gauss range of a closed convex subset of the Euclidean space changes when acting on the given set with an affine automorphism. On the other hand we investigate the spherical-convexity of the Gauss range of such a set. Since the Gauss range is not always spherically-convex, we focus our attention on the spherical-convexity of its spherical interior and of its closure as well. Examples of closed convex sets with spherically-convex Gauss ranges are, among others, the epigraphs of suitable lower semicontinuous convex functions.
Keywords: Closed convex set, Gauss range, spherically-convex set.
MSC: 52A20, 26B25.
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