12 problems

Let $V_d=\{f_{1,d}=\cdots=f_{r,d}=0\}\subset(\mathbb C^\ast)^n$ be the complete intersection under consideration, and let $\Delta_{i,d}$ be the Newton polytope of $f_{i,d}$. Set…

Let $K,L\in\mathcal{K}_e(\mathbb{R}^n)$ be origin-symmetric convex bodies containing the origin in their interiors. Let $V$ denote the mixed volume functional, let $S_K$ be the sur…

Schneider's conjecture. The support of the mixed area measure satisfies

Let $K\subset\mathbb{R}^n$ be a convex body. Let $\Delta_p:\mathbb{R}^n\to(\mathbb{R}^n)^p$ be the diagonal embedding, and let $\iota_i:\mathbb{R}^n\to(\mathbb{R}^n)^p$ be inclusio…

Let $K\subset\mathbb{R}^n$ be a convex body, and let $V$ denote mixed volume, with $K[k]$ denoting $k$ copies of $K$. Godbersen's conjecture. For each convex body…

Let $K\subset\mathbb{R}^n$ be a convex body, let $\lambda\in(0,1)$, and define the unbalanced difference body … Let $\Delta$ be an $n$-dimensional simplex. Unbalanced difference-bo…

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

Let $K\in\mathcal{K}_0^n$ be a convex body, where $\mathcal{K}_0^n$ denotes the convex bodies in $\mathbb{R}^n$ containing the origin in their interiors. Define … Here $V_n$ is the…

Betke–Weil conjecture.

Godberson's conjecture.

Let $d$ be a positive integer, let $m$ be a real number, and let $K_1,\ldots,K_d$ be convex bodies in $\mathds{R}^d$ satisfying … For an integer $\ell$ with $1\leq\ell\leq d$, the…

The local $L^p$-Brunn-Minkowski inequality. One has