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Journal of Convex Analysis, Vol. 7, No. 1, pp. 129-166 (2000)
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On the Algebraic Properties of Convex Bodies and Some Applications

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Svetoslav Markov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, "Acad. G. Bonchev" st., block 8, 1113 Sofia, Bulgaria, smarkov@iph.bio.bas.bg

**Abstract:** 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. Radstr\"{o}m 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

**Classification (MSC2000):** 52A01, 52A05, 06F20, 15A03, 65G10

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