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Journal of Convex Analysis 07 (2000), No. 1, 129--166
Copyright Heldermann Verlag 2000

On the Algebraic Properties of Convex Bodies and Some Applications

Svetoslav Markov
Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, 113 Sofia, Bulgaria

We extend the set of convex bodies up to differences (factorized pairs) of convex bodies; thereby (Minkowski) multiplication by real scalar is extended in a natural way. We show that differences of convex bodies form a special quasilinear space with group structure. The latter is abstractly studied by introducing analogues of linear combinations, dependence, basis, associated linear spaces etc. A theorem of H. Radström for embedding of convex bodies in a normed vector space is generalized. Support functions and their differences are discussed in relation to quasilinear spaces.

Keywords: Differences of convex bodies, Minkowski operations, quasilinear spaces, differences of support functions.

MSC: 52A01; 52A05, 06F20, 15A03, 65G10

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