On quasilinear spaces of convex bodies and intervals. (Q1421219)

Quasilinear spaces are axiomatic characterizations of spaces of convex bodies, i.e., of interval spaces. They show a semigroup structure with cancellation law w.r.t. (Minkowski) addition, and further, a scalar multiplication. The paper focuses on quasivector spaces which are natural extensions of quasilinear spaces. They have a group structure w.r.t. addition. Main result is that every quasivector space admits a representation as a direct sum of a vector space and a symmetric quasivector space. Finite-dimensional symmetric quasivector spaces are analysed as well and it is proven that arithmetic operations in such spaces can be reduced to equivalent operations in canonical vector spaces.