Journal of Convex Analysis 12 (2005), No. 1, 131--137
Copyright Heldermann Verlag 2005
Separability of H-convex Sets
V. Boltyanski
CIMAT, A.P. 402, 36000 Guanajuato, Gto., Mexico
boltian@cimat.mx
H. Martini
Faculty of Mathematics, Technical University, 09107 Chemnitz, Germany
martini@mathematik.tu-chemnitz.de
We consider the problem of H-separability for two H-convex subsets A and B of Rn. There are two types of H-separability. The first one, called "strict H-separability", is the separation (in the usual sense) of the sets A and B by an H-convex hyperplane. The second one ("weak H-separability") means to look for an H-convex half-space P such that A is situated in P, whereas B has no point in common with the interior of P. We give necessary and sufficient conditions for both these types of H-separability; the results are connected to the H-convexity of the Minkowski sum of H-convex sets investigated in a previous paper of the authors [J. Combin. Theory, Ser. A 103 (2003) 323--336]. Some examples illustrate the obtained results.
Keywords: Convex sets, H-convexity, Minkowski addition, separation theory.
MSC: 52A01; 52A20
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