On the Gauss Range of a Closed Convex Set

Juan Enrique Martínez-Legaz; Cornel Pintea

doi:10.68381/jca30054

Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Juan Enrique Martínez-Legaz

Dep. d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, Barcelona Graduate School of Mathematics, Spain
and: MOVE (Markets, Organizations and Votes in Economics)

JuanEnrique.Martinez.Legaz@uab.es

Cornel Pintea

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

cpintea@math.ubbcluj.ro

Keywords

Closed convex set
Gauss range
spherically-convex set

Mathematics Subject Classification

52A20, 26B25

References

O. P. Ferreira, A. N. Iusem, S. Z. Németh: Projections onto convex sets on the sphere, J. Global Optim. 57 (2013) 663–676.https://doi.org/10.1007/s10898-012-9914-3 Google Scholar
M. A. Goberna, E. González, J. E. Martínez-Legaz, M. I. Todorov: Motzkin decomposition of closed convex sets, J. Math. Analysis Appl. 364 (2010) 209–221.https://doi.org/10.1016/j.jmaa.2009.10.015 Google Scholar
M. A. Goberna, J. E. Martínez-Legaz, M. I. Todorov: On Motzkin decomposable sets and functions, J. Math. Analysis Appl. 372 (2010) 525–537.https://doi.org/10.1016/j.jmaa.2010.07.007 Google Scholar
J. E. Martínez-Legaz, C. Pintea: Closed convex sets with an open or closed Gauss range, Math. Program. Ser. B 189/1-2 (2021) 433–454.https://doi.org/10.1007/s10107-020-01561-7 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar
R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.

[1] O. P. Ferreira, A. N. Iusem, S. Z. Németh: Projections onto convex sets on the sphere, J. Global Optim. 57 (2013) 663–676.https://doi.org/10.1007/s10898-012-9914-3 Google Scholar

[2] M. A. Goberna, E. González, J. E. Martínez-Legaz, M. I. Todorov: Motzkin decomposition of closed convex sets, J. Math. Analysis Appl. 364 (2010) 209–221.https://doi.org/10.1016/j.jmaa.2009.10.015 Google Scholar

[3] M. A. Goberna, J. E. Martínez-Legaz, M. I. Todorov: On Motzkin decomposable sets and functions, J. Math. Analysis Appl. 372 (2010) 525–537.https://doi.org/10.1016/j.jmaa.2010.07.007 Google Scholar

[4] J. E. Martínez-Legaz, C. Pintea: Closed convex sets with an open or closed Gauss range, Math. Program. Ser. B 189/1-2 (2021) 433–454.https://doi.org/10.1007/s10107-020-01561-7 Google Scholar

[5] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar

[6] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar