The $p$-cones in dimension $n \geq 3$ are not homogeneous when $p\neq 2$

Abstract

Using the T-algebra machinery we show that, up to linear isomorphism, the only strictly convex homogeneous cones in $\Re^n$ with $n \geq 3$ are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when $p\neq 2$, $1 < p <\infty$ and $n\geq 3$, thus answering a problem proposed by Gowda and Trott.