We use optimal transport and equivariant topology to show the next Borsuk Ulam/Ham sandwich type statement. Given a prime p smaller or equal than the dimension, an absolutely continuous probability measure m, a convex body K, and a continuous functional F from the space of convex bodies to the real numbers. It is always possible to partition the convex body K into p convex pieces K_1,K_2...K_p, such that m(K_1)=m(K_2)=...m(K_p)=1/p and F(K_1)=F(K_2)=...F(K_p). Simultaneously This is joint work with B.Aronov.