32 problems

Let $K$ and $L$ be convex bodies in $\mathbb{R}^n$. For $\lambda\in(0,1)$, their geometric mean is defined, for origin-symmetric bodies, by … Here $V$ denotes volume,…

Optimal partial plank covering conjecture. There exists a plank $P$ of width $W$ such that, for every such finite family,

Let $K$ be a convex body in $\mathbb{R}^n$ and let $(\sigma_1,\dots,\sigma_m)$ be an $s$-cover of $\sigma\subset[n]$. For each $\tau\subset[n]$, let $H_\tau$ denote the coordi…

The volume inequality. One has

Let $T$ be a tree, meaning a connected circuit-free non-oriented finite simple graph, and let $C_T(\mathbb{R}^3)$ be its configuration space. Let $\mathcal{A}_T$ be the associated…

Let $\Omega$ be a strictly convex domain with smooth boundary $M$ in $\mathbb{S}^{n+1}$. For integers $k$ and $l$ satisfying $0\leqslant k\leqslant n$ and $0\leqslant l\leqslant k$…

Let $M$ be a compact hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. Write…

Let $\mathcal{K}^2$ denote the class of planar convex sets, and let $\omega(\Omega)$, $h(\Omega)$, and $R(\Omega)$ denote respectively the minimal width, Cheeger constant, and circ…

Let $\mathcal{K}^2$ denote the class of planar convex sets, and let $\omega(\Omega)$, $h(\Omega)$, and $P(\Omega)$ denote respectively the minimal width, Cheeger constant, and peri…

Higher-mean-curvature hyperbolic Michael–Simon conjecture. For $1\le k\le n$, the inequality

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. For $1\leq k\leq n-1$, write…

Weighted quermassintegral conjecture. One has

Let $\mathbb{B}^2$ be the unit disk, and let $s_{\mathbb{B}^2}$ and $p_{\mathbb{B}^2}$ denote the triangular ratio and point-pair metrics, respectively. For $x,y\in\mathbb{B}^2$, l…

Let $n>k\geq 1$, let $\Box^n=[-1,1]^n$, and let $H$ be a $k$-dimensional linear subspace. Write $C_{\Box}(n,k)2^k$ for the maximum volume among sections $\Box^n\cap H$ that are aff…

Let $\Sigma\subset\mathbb{R}^3$ be an immersed connected closed surface. Topping's conjecture. … This conjecture asks for the optimal lower bound for the total absolute mean curvat…

Let $M^n$ be a compact Riemannian manifold with boundary $\Sigma=\partial M$, $Ric\geq n-1$, and second fundamental form $\Pi\geq 0$. Boundary area rigidity conjecture. … Moreover,…

Let $M^n$ be a compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on $\partial M$. Boundary area conjecture. … The text presents this as a consequ…

Centrally symmetric reverse isodiametric inequality.

Optimal reverse isodiametric inequality.

Alexandrov-Fenchel inequality. One should have

Generalized Busemann intersection inequality. More generally, for an arbitrary real number $m$ such that the integral on the right-hand side exists in the Lebesgue sense,

Saroglou's lower-order projection-body conjecture. For $i=1,\dots,d-2$, the functionals $P_i(\cdot)$ and $Q_i(\cdot)$ are minimized precisely for Euclidean balls. The source notes…

Let $K$ be a $d$-dimensional convex body containing the origin in its interior. Define … For $x\in S^{d-1}$, let $K\cap(sx+x^\bot)$ be the $(d-1)$-dimensional section and let…

Let $K$ be a symmetric convex body in $\mathbb{R}^3$, let $h_K$ be its support function, let $V(K)$ be its volume, and let $V_2$ denote planar area. For $x\in S^2$, write…

Let $(M,g,k)$ be a simply connected, axially symmetric, maximal initial data set with multiple ends, one $(M_{\text{end}}^{1})$ asymptotically flat and the others…