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

In this paper, we determine the automorphism group of the $p$-cones ($p\neq 2$) 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.