18 problems

Conjecture on ultra-log-concave intrinsic-volume sequences. The sequence $(a_k)_{k\geq 0}$ is the intrinsic-volume sequence of some infinite-dimensional GB-set if and only if

Gao–Vitale conjecture. Either

Let $r>r_0>0$, $N>1$, $d>k>0$, and let $P=\{\mathbf p_1,\dots,\mathbf p_N\}\subset\mathbb{E}^d$ satisfy $r_{cr}(P)=r_0$, where $r_{cr}$ is the circumradius. Let…

Let $d>1$, $r>0$, and $0<k\leq d-2$. An $r$-ball body is an intersection of finitely many balls of radius $r$ in $\mathbb{E}^d$; its inradius is the radius of the largest ball cont…

Let $L_1$ denote the space of integrable functions, and let $K\subseteq L_1$ be compact and convex. Write $\mu_1^{L_1}(K)$ for the supremum of the first Holmes–Thompson intrinsic v…

Let $X\subset\mathbb{R}^n$ be compact and convex, and let $\mathcal{M}_X(R):=\operatorname{mag}(X,R\cdot\mathrm{d})$ for $R>0$. For $0\leq k\leq n$, let $V_k(X)$ denote the intrins…

Variant of the Leinster–Willerton conjecture. There are universal constants $\gamma_{0,n},\gamma_{1,n},\ldots,\gamma_{n,n}$ such that, for every such $X$,

Let $P\in\mathcal P(\mathbb{Z}^d)$ have dimension $r$, and let the $k$-th discrete intrinsic volume be represented in the binomial basis by … A valuation is combinatorially positiv…

Betke–Weil conjecture.

Let $\mathcal E_1$ and $\mathcal E_2$ be two ellipsoids in $\mathbb R^n$. For an ellipsoid $\mathcal E$, let $V_i(\mathcal E)$ denote its $i$-th intrinsic volume. Gusakova–Zaporozh…

Let $V_j$ be finite-dimensional real vector spaces equipped with smooth norms $F_j$, and let $e_j\colon M\to V_j$, $j=1,2$, be smooth immersions of a smooth manifold $M$. For a val…

Let ${\mathbb E}^d$ be Euclidean $d$-space, let $P=\{\mathbf{p}_1,\mathbf{p}_2,\dots,\mathbf{p}_N\}\subseteq{\mathbb E}^d$, and let $P^r$ denote the intersection of the closed ball…

Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be ellipsoids in $\mathbb{R}^n$. For a bounded convex set $K\subset\mathbb{R}^n$, let $V_0(K),\ldots,V_n(K)$ denote its intrinsic volumes, d…

Let $X$ be a compact weakly smoothable Alexandrov space, meaning that it is a Gromov–Hausdorff limit of closed smooth connected Riemannian manifolds with uniformly bounded above di…

Let $\{M_l^n\}_{l=1}^{\infty}$ be smooth $n$-dimensional closed connected Riemannian manifolds with a uniform lower bound on sectional curvature, converging in the Gromov–Hausdorff…

Let $n\in\mathbb{N}$, $D>0$, and $\kappa\in\mathbb{R}$. For a closed smooth connected Riemannian manifold $(M^n,g)$, write $V_i(M)$ for its intrinsic volumes. The boundedness conje…

Let $A$ be a compact $\ell_1$-convex set, meaning a geodesic subset of $\ell_1^n$, and let $V'_i(A)$ denote its $i$th $\ell_1$-intrinsic volume. These volumes are defined by the St…

Let $\operatorname{Sym}^k_+$ denote the cone of positive semidefinite matrices, and let $e(C)$ be the excess of a closed convex cone $C$ over the Lorentz cone. Positive-semidefinit…