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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 MartŪnez-Legaz
Dep. d'Economia i d'HistÚria EconÚmica, Universitat AutÚnoma de Barcelona, Barcelona Graduate School of Mathematics, Spain

Cornel Pintea
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania, Romania

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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