On a generalization of the convex addition

We study an operation of generalized addition of convex bodies ($p$-addition, $p\in[1,+\infty]$), converting the class of bounded symmetrical convex subsets of a normed linear space into an Abelian subgroup with a natural action of positive scalars on it. We investigate properties of this operation and mention its applications to the problem of computing of the joint spectral radius of linear operators. We prove that $p$-addition is the unique associative operation among some natural class of binary operations on the set of symmetrical convex bodies.