Hard Lefschetz and Hodge–Riemann conjecture for smooth valuations

Let $n$ be a positive integer, let $0\leq r\leq\lfloor\frac n2\rfloor$, and let $\mathcal K(\mathbb R^n)$ denote the space of convex bodies in $\mathbb R^n$. For $i=0,\dots,n-2r$, choose $C_i\in\mathcal K(\mathbb R^n)$ with smooth and strictly positively curved boundary, and define the valuation $\mu_{C_i}$ by

$$\mu_{C_i}(K)=V(K,\dots,K,C_i),\qquad K\in\mathcal K(\mathbb R^n).$$

Let $\operatorname{Val}_k^\infty(\mathbb R^n)$ denote the space of smooth translation-invariant valuations of degree $k$, let $*$ denote convolution, and let $\overline{\psi}$ denote complex conjugation. Hard Lefschetz and Hodge–Riemann conjecture. The map

$$\operatorname{Val}_{n-r}^\infty(\mathbb R^n)\longrightarrow\operatorname{Val}_r^\infty(\mathbb R^n),\qquad \phi\longmapsto\phi*\mu_{C_1}*\cdots*\mu_{C_{n-2r}}$$

is an isomorphism of topological vector spaces, and the sesquilinear form

$$(\phi,\psi)\longmapsto(-1)^r\,\phi*\overline{\psi}*\mu_{C_1}*\cdots*\mu_{C_{n-2r}}$$

is positive definite on

$$\left\{\phi\in\operatorname{Val}_{n-r}^\infty(\mathbb R^n)\mid\phi*\mu_{C_0}*\cdots*\mu_{C_{n-2r}}=0\right\}.$$

These properties are the valuation-theoretic analogues of the hard Lefschetz theorem and Hodge–Riemann relations for convex-geometric intersection structures. The source presents them as the main problem motivating the paper; no resolution is supplied in the given context.