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Journal of Convex Analysis 27 (2020), No. 2, 407--421
Copyright Heldermann Verlag 2020

On Far and Near Ends of Closed and Convex Sets

Minghua Li
School of Mathematics and Finance, Chongqing University of Arts and Sciences, Yongchuan, Chongqing 402160, China

Kaiwen Meng
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Xiaoqi Yang
Dept. of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

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.-W. Meng, V. Roshchina and X.-Q. Yang [On local coincidence of a convex set and its tangent cone, J. Optim. Theory Appl. 164 (2015) 123--137] 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.

Keywords: Closed and convex sets, far ends, near ends, faces, exposed faces, support functions, gauge, cogauge.

MSC: 52A10, 52A15, 52A20, 52A99.

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