platonic solids and surfaces of constant norm

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?

5 thoughts on “platonic solids and surfaces of constant norm

  1. If I remember correctly, given any convex solid which is (reflectively) symmetric about the origin you can create a norm by simply taking every direction and defining a unit vector in that direction by the surface of the solid. The convexity should make the triangle inequality hold. But I might be totally wrong.

    PS – Happy birthday.

  2. Wow.

    I didn’t want to hear it this way, Anand.

    I guess I thought that, after everything they’ve been through, the solids would be something more than “platonic”.

    “Platonic” is hardly the word I would use to decribe what happened on New Year’s Eve.

    I guess some solids’ definitions of friendship are different than others.

    I should have known. I didn’t listen to you when you told me that the sphere exists in “every dimension”. I guess one isn’t enough by some standards.

  3. Oh – and by the way – Norm is anything but constant. That bastard has been in and out of my life – and his children’s – since 1996.

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