Warning : this is all tenuous connections in my head and probably has some horrible misunderstanding in it.

In high dimensions (> 4) there are only 3 regular polyhedra — the n-simplex, n-ocahedron, and n-cube. The latter are also surfaces of constant l₁-norm and l_∞-norm respectively. Those two can be thought of as “dual” to each other since 1/1 + 1/∞ = 1 (by analogy with p norms). The sphere is also a regular body that exists in every dimension, and is a surface of constant 2-norm, and 2 is self-dual. Is there some crazy norm or metric such that the simplex is also a surface of constant norm?