Li, Minghua; Meng, Kaiwen; Yang, Xiaoqi On far and near ends of closed and convex sets. (English) Zbl 1439.52004 J. Convex Anal. 27, No. 2, 407-421 (2020). Summary: We first introduce for a closed and convex set two classes of subsets: the near and far ends relative to a point, and give some full characterizations for these end sets by virtue of the face theory of closed and convex sets. We also provide some connections between closedness of the far (near) end and the relative continuity of the gauge (cogauge) for closed and convex sets. Moreover, motivated by these connections, we clarify Conjecture 6.1 of K. Meng et al. [J. Optim. Theory Appl. 164, No. 1, 123–137 (2015; Zbl 1309.49013)] in the sense that the gauge of a closed and convex set containing 0 is continuous relative to its domain if it is relatively continuous at 0, holds always in the two-dimensional case, but is not true in general by constructing a three-dimensional counterexample. MSC: 52A10 Convex sets in \(2\) dimensions (including convex curves) 52A15 Convex sets in \(3\) dimensions (including convex surfaces) 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) 52A99 General convexity 49J52 Nonsmooth analysis 49J53 Set-valued and variational analysis Keywords:closed and convex sets; far ends; near ends; faces; exposed faces; support functions; gauge; cogauge Citations:Zbl 1309.49013 PDFBibTeX XMLCite \textit{M. Li} et al., J. Convex Anal. 27, No. 2, 407--421 (2020; Zbl 1439.52004) Full Text: Link References: [1] A. Barbara, J.-P. Crouzeix:Concave gauge functions and applications, Z. Oper. Res. 40 (1994) 43-74. · Zbl 0810.90102 [2] M. J. Cánovas, R. Henrion, M. A. López, J. Parra:Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming, J. Optim. Theory Appl. 169 (2016) 925-952. · Zbl 1342.90198 [3] M. J. Cánovas, M. A. López, J. Parra, F. J. Toledo:Calmness of the feasible set mapping for linear inequality systems, Set-Valued Var. Analysis 22 (2014) 375- 389. · Zbl 1297.90163 [4] G. Cornuéjols, C. Lemaréchal:A convex-analysis perspective on disjunctive cuts, Math. Program. 106 (2006) 567-586. · Zbl 1149.90175 [5] A. Eberhard, V. Roshchina, T. Sang:Outer limits of subdifferentials for min-max type functions, Optimization 68(7) (2019) 1391-1409. · Zbl 1461.90145 [6] H. Hu:Characterizations of the strong basic constraint qualifications, Math. Oper. Res. 30 (2005) 956-965. · Zbl 1278.90310 [7] H. Hu:Characterizations of local and global error bounds for convex inequalities in Banach spaces, SIAM J. Optimization 18 (2007) 309-321. · Zbl 1176.90642 [8] H. Hu, Q. Wang:Closedness of a convex cone and application by means of the end set of a convex set, J. Optim. Theory Appl. 150 (2011) 52-64. · Zbl 1229.52004 [9] A. D. Ioffe:Metric regularity – a survey. Part 1: Theory, J. Aust. Math. Soc. 101 (2016) 188-243. · Zbl 1369.49001 [10] M.-H. Li, K.-W. Meng, X.-Q. Yang:On error bound moduli for locally Lipschitz and regular functions, Math. Program. Ser. A 171 (2018) 463-487. · Zbl 1397.65095 [11] R. Lucchetti:Convexity and Well-Posed Problems, Springer, Cham (2006). · Zbl 1106.49001 [12] K.-W. Meng, V. Roshchina, X.-Q. Yang:On local coincidence of a convex set and its tangent cone, J. Optim. Theory Appl. 164 (2015) 123-137. · Zbl 1309.49013 [13] J.-P. Penot, C. Zalinescu:Harmonic sum and duality, J. Convex Analysis 7 (2000) 95-113. · Zbl 0964.49020 [14] R. T. Rockafellar:Convex Analysis, Princeton University Press, Princeton (1970). · Zbl 0193.18401 [15] R. T. Rockafellar, R. J.-B. Wets:Variational Analysis, Springer, Berlin (1998). [16] A. M. Rubinov:Radiant sets and their gauges, in:Quasidifferentiability and Related Topics, Nonconvex Optim. Appl. 43 (2000) 235-261. · Zbl 0990.90131 [17] A. Zaffaroni:Convex coradiant sets with a continuous concave cogauge, J. Convex Analysis 15(2) (2008) 325-343. · Zbl 1148.52002 [18] X. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.