The automorphism group and the non-self-duality of p-cones

Abstract

In this paper, we determine the automorphism group of the p-cones (p = 2) in dimension greater than two. In particular, we show that the automorphism group of those p-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the p-cones. Under the Euclidean inner product it is well-known that a p-cone is self-dual only when p = 2. However, it was not known whether it is possible to construct an inner product depending on p which makes the p-cone self-dual. Our results shows that no matter which inner product is considered, a p-cone will never become self-dual unless p = 2 or the dimension is less than three.

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

“…These issues were later settled in [13,14] using techniques such as T-algebras [33] and tools borrowed from differential geometry. In this final subsection, we show "easy" proofs for the questions above based on our error bound results.…”

“…The other p-cones are not symmetric and do not typically have closed form expressions for their projections. See [13,14] for a discussion on the extent to which they fail to be symmetric.…”