This is from Garding.

Let $P\in{\mathbb R}[X_1,\ldots,X_d]$ be a homogeneous polynomial. Assume that it is hyperbolic in some direction $e\in{\mathbb R}^d$ (with say the normalisation $P(e)=1$) and let $\Gamma$ be its cone of future, that is the connected component of $e$ in the complement of $\{P=0\}$. It is known that $\Gamma$ is convex. Then we have the inverse Hölder inequality: for every $v_1,\ldots,v_n\in\Gamma$, $$M(v_1,\ldots,v_n)\ge(P(v_1)\cdots P(v_n))^{\frac1n},$$ where $M$ is the symmetric multiplinear form such taht $M(x,\ldots,x)=P(x)$.

Consequences occur in convex geometry, combinatorics, PDEs, ...

As a matter of fact, $P^{\frac1n}$ is concave over $\Gamma$. A simple example is that of quadratic forms of signature $(1,d-1)$. Another nice example is $P=\det$, where ${\mathbb R}^d={\bf M}_n({\mathbb R})$.