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1. arXiv:2207.11973 (Published 2022-07-25)

   Blaschke, Separation Theorems and some Topological Properties for Orthogonally Convex Sets

   Phan Thanh An, Nguyen Thi Le
   Comments: 17 pages, 10 figures

   Categories: math.OC, cs.CG

   Subjects: 26B25, 32F27, 52A10, 52A30

   Keywords: orthogonally convex set, topological properties, separation theorems, blaschke-type theorem, orthogonally convex paths
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   In this paper, we deal with analytic and geometric properties of orthogonally convex sets. We establish a Blaschke-type theorem for path-connected and orthogonally convex sets in the plane using orthogonally convex paths. The separation of these sets is established using suitable grids. Consequently, a closed and orthogonally convex set is represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonally convex sets in dimensional spaces are also given.

2. arXiv:1510.04487 (Published 2015-10-15)

   Some remarks on convex analysis in topological groups

   Jonathan M. Borwein, Ohad Giladi
   Categories: math.OC, math.FA

   Keywords: convex analysis, topological groups, krein-milman type theorems, separation theorems
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   We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.

3. arXiv:math/0212294 (Published 2002-12-20, updated 2003-09-29)

   Duality and separation theorems in idempotent semimodules

   Guy Cohen, Stephane Gaubert, Jean-Pierre Quadrat
   Comments: 24 pages, 5 Postscript figures, revised (v2)

   Journal: Linear Algebra and its Applications, Volume 379, pages 395--422, March 2004.

   DOI: 10.1016/j.laa.2003.08.010

   Categories: math.FA, math.OC

   Subjects: 46A20, 06F07, 46A55

   Keywords: idempotent semimodules, separation theorems, subsemimodule, corollary duality results, convex subsets

   Tags: journal article
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   We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.