40 problems

Central limit conjecture. There exists a sequence of positive numbers $(\sigma_n^2)_{n\ge1}$ such that

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…

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 $K$ be a convex body in $\mathbb{R}^n$, let $i\in\\{1,\dots,n-1\\}$, and let $\Pi_i K$ be its projection body of order $i$, defined by … where $V_i$ is the $i$th intrinsic volu…

Let $K$ be an infinite-dimensional convex body with $V_1<\infty$, and define … The sequence $m_k$ is decreasing. Extremal decay conjecture. Either … or … The Brownian-motion body…

Let $K$ be a convex set in ${\bf R}^n$, and let $a_1\leq a_2\leq\cdots\leq a_n$ be the real parts of the roots of … Let $r$ and $R$ be the radii of $K$ relative to the $n$-dimensio…

Let $(K_n)_{n\ge1}$ be a sufficiently regular sequence of convex bodies such that $K_n\subseteq \mathbb R^n$ and $\dim K_n=n$ for every $n\ge1$, and assume that … as $n\to\infty$.…

Let $d\geq 1$, let $\beta>\frac d2$, and let $\widetilde K_n^\beta$ be the random beta-prime polytope in $\mathbb{R}^d$. For $s\in\{1,\ldots,d\}$, write $V_s(\widetilde K_n^\beta)$…

Bezdek's conjecture. A $\lambda$-convex spindle has the largest circumradius among all $\lambda$-convex bodies in $\mathbb{R}^n$ with fixed mean width.

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 $d>1$, $r>0$, 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 $X$ be a compact set of $\mathbb{R}^d$ that is subanalytic. The quantities $V_i^{\varepsilon}(Y)$ are defined for compact sets $Y$ and parameter $\varepsilon>0$, and the preced…

Let $\mathcal{K}_j^n$ denote the class of convex bodies of dimension $j$ in $\mathbb{R}^n$, equipped with the Hausdorff metric $d_H$ and the intrinsic volume metric $\delta_j$. Int…

Alesker's conjecture. Under these hypotheses, the asserted existence and, in the collapsing case, the stated stratified expression for the limit of each intrinsic volume $V_i(M_j)$…

Let $\mathcal{K}^n$ denote the compact convex bodies in $\mathbb{R}^n$, let $\mathbb{Z}^n$ be the integer lattice, and let the Wills functional be … where $V_m(K)$ are the Euclidea…

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…

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 $P:=\{\mathbf{p}_1,\dots,\mathbf{p}_N\}$ and $Q:=\{\mathbf{q}_1,\dots,\mathbf{q}_N\}$ be finite point sets in the Euclidean plane ${\mathbb E}^2$, with $Q$ a contraction of $P$…