From John to Gauss–John positions via dual mixed volumes

We investigate the optimization of the dual mixed volumes where K⊆Rn is a convex body, Dn the Euclidean ball and SK runs over all positions of K. When S is linear we give necessary and sufficient conditions for K to be in extremal position in terms of a decomposition of the identity. We consider affine problems and we also present an approach involving parallel sections of K which can be understood as a dual fractional Kubota formula.