Journal of Convex Analysis, Vol. 7, No. 1, pp. 129-166 (2000)

On the Algebraic Properties of Convex Bodies and Some Applications

Svetoslav Markov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, "Acad. G. Bonchev" st., block 8, 1113 Sofia, Bulgaria,

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

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number]
© 2000 ELibM for the EMIS Electronic Edition