  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.
