On the algebraic properties of convex bodies and related algebraic systems

The algebraic system of convex bodies with addition and multiplication by scalar is studied. A new operation for convex bodies, called inner addition, is introduced. New distributivity relations for xonvex bodies, called resp. quasidistributive and $q$-distributive law, are formulated and proved. The convex bodies form a quasilinear system with respect to addition and multiplication by scalar. The latter is isomorphically embedded in a $q$-linear system, which is an abelian group with respect to addition and obeys the $q$-distributive law. A result of H. Rådström for convex bodies is generalized.