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. 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.

Author information

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Svetoslav Markov

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

Keywords

  • Differences of convex bodies
  • Minkowski operations
  • quasilinear spaces
  • differences of support functions

Mathematics Subject Classification

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

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Suggested citation

S. Markov. “On the Algebraic Properties of Convex Bodies and Some Applications.” Journal of Convex Analysis 7 (2000), No. 1, 129–166. https://doi.org/10.68381/jca07007

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Published by Heldermann Verlag, 2000. Rights now held by Banach Press LLC.