Convex body  In mathematics, a convex body in "n"-dimensional Euclidean space formula_1 is a compact convex set with non-empty interior. A convex body "K" is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point "x" lies in "K" if and only if its antipode, −"x", also lies in "K". Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on R"n". Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.