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

## TL;DR

This paper proves that p-cones are not homogeneous in dimensions three and higher unless p equals 2, using T-algebra machinery to classify strictly convex homogeneous cones.

## Contribution

It demonstrates that the only strictly convex homogeneous cones in higher dimensions are Lorentz cones, resolving a previously open problem about p-cones.

## Key findings

- p-cones are not homogeneous for p ≠ 2 in dimensions n ≥ 3
- Lorentz cones are the only strictly convex homogeneous cones in these dimensions
- The result answers a question posed by Gowda and Trott

## 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.