Hodge–Riemann and equality conjecture for mixed-volume valuations

Let $E_i=(K_1^{(i)},\ldots,K_{t_i}^{(i)})$, $1\leq i\leq p$, be $p$ tuples of convex bodies, and let $\lambda^1,\ldots,\lambda^p$ be partitions satisfying

$$\sum_{i=1}^p |\lambda^i|=n-2.$$

For a tuple $E_i$, let $s_{\lambda^i}(E_i)$ denote the corresponding Schur mixed-volume expression, and define

$$\Theta(M,N)=V\bigl(s_{\lambda^1}(E_1),\ldots,s_{\lambda^p}(E_p),M,N\bigr).$$

Hodge–Riemann and equality conjecture. If all convex bodies in the tuples $E_i$ are smooth and strictly convex, then the valuation

$$\Theta=V\bigl(s_{\lambda^1}(E_1),\ldots,s_{\lambda^p}(E_p),-,-\bigr)$$

satisfies the Hodge–Riemann relation: for a smooth convex body $K$ with nonempty interior and every smooth valuation $\phi\in\operatorname{Val}_{n-1}^\infty$ satisfying

$$\Theta*V(K;-)*\phi=0,$$

one has

$$\Theta*\phi*\phi\leq 0,$$

with equality if and only if $\phi=0$. If all convex bodies in the tuples $E_i$ are smooth and have nonempty interior, then for any two convex bodies $M,N$,

$$\Theta(M,N)^2=\Theta(M)\Theta(N)$$

if and only if $M$ and $N$ are homothetic. The conjecture proposes a Hodge–Riemann-type strengthening of Alexandrov–Fenchel inequalities for mixed volumes and characterizes equality by homothety.