18 problems

Let $K$ be a compact subset of ${\mathbb R}^d$. Choose directions independently and uniformly at random from the unit sphere, and apply the corresponding sequence of Steiner symmet…

Let $U$ and $R$ be domains in the setting of Theorem, let $z$ be the specified starting point, and write $\tau_U$, $\tau_R$, $\tau_{U^+}$, and $\tau_{R^+}$ for the corresponding fi…

Rate-of-convergence conjecture. There exist $c n\log\frac{1}{\varepsilon}$ Steiner symmetrizations sufficient to transform $K$ into such a body $\widetilde{K}$, where $c>0$ is a nu…

Let $X$ have the exponential distribution, specifically $X\sim\operatorname{Exp}(1)$ as in the surrounding discussion, and consider Variance symmetrization resistance. Exponential…

Let $X$ have the exponential distribution on $\a0\mathbb{R}_+$, and let $V$- and $h$-symmetrization resistance be considered in the space $\mathcal{P}_c(\mathbb{R})$ of compactly s…

Let $U$ be a bounded simply connected domain containing the origin. Let $Z_t$ be a standard planar Brownian motion run inside $U$, let $\tau_U$ be its exit time, and let $\mu_U$ be…

Let $\Omega$ be a bounded domain and let $H$ be a polarizer. The reverse Faber–Krahn conjecture. … The inequality is immediate in the cases discussed in the source when the transfi…

Let $E$subset ${\mathbb R}^2$ be compact and let $H$ be a polarizer. The polarization monotonicity conjecture. … Moreover, equality holds only if $P_H(E)\cong E$ or…

Let $U$ be a planar domain, let $\mathfrak{B}(U)$ denote its Brownian symmetrization, and let $\lambda_1(U)$, $|U|$, and $\operatorname{Per}(U)$ denote respectively the principal D…

Consider the planar improper partitioning problem with one finite-measure chamber $\Omega_0$ of prescribed area and two infinite-measure chambers $\Omega_+$ and $\Omega_-$. Compact…

Let $({\mathcal L}^0,\mathbf e)$ be nondegenerate and strictly admissible. For generic admissible $({\mathcal L},\mathbf e)$ satisfying the partial symmetry hypothesis, with…

Let $({\mathcal L},{\mathbf e})$ satisfy the appropriate nondegeneracy and admissibility hypotheses, and suppose that $\mathcal L$ satisfies the partial symmetry hypothesis. A maxi…

Let $(B,\|\cdot\|)$ be a Banach space, let $X_1,\ldots,X_n\in B$ be independent identically distributed random variables with law $\mu$, and let…

Let $K$ be a convex body in $\mathbb{R}^n$, and let $S_uK$ denote Steiner symmetrization in the direction $u$. Repeated Steiner symmetrizations approximate a Euclidean ball, with t…

Let $g(\cdot,0)$ and $g^{*}(\cdot,0)$ be the Green functions of $\Omega_{\alpha}$ and $\Omega_{\widetilde{\alpha}}$, respectively, both with pole at the origin. Let…

Let $G$ be an $n$-vertex graph with $e$ edges. Write ${\rm Tr}(G)$ for the number of edges of $G$ contained in triangles, and let ${\rm Tr}(n,e)$ be the minimum of ${\rm Tr}(G)$ ov…

Let $B$ be a ball in $R^N$, and let $lambda^D_1(Omega)$ denote the first eigenvalue of the Dirichlet Laplacian on $Omega$. For domains $Omega\tsubset R^N$ with $|Omega|=|B|$, Rayle…

In $\mathbb{R}^{N}$, let the density be smooth, radial, and log-convex. Bayle's conjecture. Balls centered at the origin are isoperimetric. The conjecture is motivated by a propose…