A lower bound is obtained for the Lipschitz extension modulus of a finite dimensional normed space in the form of a universal power of its dimension. The core technical contribution is a geometric structural result on stochastic clustering of finite dimensional normed spaces, which implies upper bounds on their Lipschitz extension moduli. The upper and lower bounds on the separation moduli of finite dimensional normed spaces obtained here relate them to well-studied volumetric invariants.

Using these connections, the asymptotic growth rate of the separation moduli of various normed spaces are determined. In the presence of enough symmetries, these bounds imply that the separation modulus is equal, up-to factors of lower order, to the product of the volume ratio of the unit ball of the dual and the square root of its dimension. A conjecture is formulated on isomorphic reverse isoperimetric properties of symmetric convex bodies that can be used with the volumetric bounds to obtain many more exact asymptotic evaluations of the separation moduli of normed spaces. The estimates on the separation modulus imply asymptotically improved upper bounds on the Lipschitz extension moduli of various classical spaces.