Dual cone and polar cone

If X is a topological vector space over the real or complex numbers, then the dual cone of a subset C ⊆ X is the following set of continuous linear functionals on X: which is the polar of the set -C.[1] No matter what C is,

Alternatively, many authors define the dual cone in the context of a real Hilbert space (such as Rn equipped with the Euclidean inner product) to be what is sometimes called the internal dual cone.

This is slightly different from the above definition, which permits a change of inner product.

For instance, the above definition makes a cone in Rn with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in Rn is equal to its internal dual.

So are all cones in R3 whose base is the convex hull of a regular polygon with an odd number of vertices.