39 problems

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 $\widehat{\Sigma}_i\in\mathcal{K}_{\omega_0}$ for $1\leqslant i\leqslant n+1$, where $\mathcal{K}_{\omega_0}$ is the class of anisotropic $\omega_0$-capillary convex bodies, an…

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

McMullen's conjecture. Mixed volumes span a dense subspace of $\operatorname{Val}(\mathbb{R}^n)$.

Let $A$subset be a convex body in $\mathbb{R}^n$, and let $2\leq r\leq n$. For convex bodies $B_1,\dots,B_r$, write $V(B_1,\dots,B_r,A[n-r])$ for their mixed volume, with $A$ repea…

Real-frequency extension conjecture. The Gelfond–Khovanskii formula for the mean value of an exponential sum over the zeros of a system holds for exponential sums with real frequen…

Let $n\geq 2$, and consider the system of equations in $(v_1,\ldots,v_n,\lambda_1,\lambda_2)\in(\mathbb{C}^{*})^{n+2}$ given by the gradient and constraint equations in the paper.…

Let $K\subset\mathbb{R}^n$ be a convex body, let $V(K[k],-K[n-k])$ denote the mixed volume with $k$ copies of $K$ and $n-k$ copies of $-K$, and let $\operatorname{vol}_n(K)$ denote…

For convex bodies $A,B$ in $\mathbb{R}^4$, let $A\wedge B$ denote the vector of mixed volumes of their coordinate projections, and let $\Delta(\mathbb{T}_2)$ be the cone appearing…

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…

McMullen's conjecture. Every monotone translation invariant valuation that is homogeneous of any order $p$ is given by a mixed volume.

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…

Let $f_{A_1,A_2,A_3}$ be the volume polynomial associated with three convex bodies $A_1,A_2,A_3$ in dimension three. A homogeneous polynomial is strongly log-concave when it has th…

Three-dimensional Blaschke–Lebesgue conjecture. The three-dimensional analogue of the Reuleaux triangle is the constant-width shape minimizing the volume.

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…

Improved concavity conjecture. The function $P^{1/m}(t)$ should be concave for $t\geq 0$.

Let $A,L_1,\ldots,L_j$ be convex bodies in $\mathbb{R}^n$, and let $V(\cdot)$ denote mixed volume; repeated entries are indicated by brackets, so $A[n-j]$ means that $A$ occurs…

Let $K,L\in\mathcal{Z}(\mathbb{R}^m)$ be zonoids and let $C\in\mathcal{G}(m-2,\mathbb{R}^m)$ be a Grassmannian zonoid of degree $m-2$. Conjecture. For any…

Let $\mathcal{P}(\Delta)$ denote the class of simple strongly isomorphic polytopes associated with a simple polytope $\Delta$, and let $C^\infty_+$ denote the class of convex bodie…

Fedotov's conjecture. One has

Betke–Weil conjecture.

Let $C_1,\ldots,C_{n-2}$ be general convex bodies rather than polytopes, and consider the extremal characterizations established in the paper for the Alexandrov–Fenchel inequality.…