250 problems

A Cartan-Hadamard manifold $M^n$ is a complete simply connected Riemannian manifold of nonpositive sectional curvature. For a bounded region $a9\subset M^n$ of finite perimeter, wr…

Petty's projection conjecture. The quantity $\mathcal{F}(K)$ is minimized over all convex bodies $K$ in $\mathbb{R}^n$ if and only if $K$ is an ellipsoid.

Let $\mathbb{R}^{n}$ be equipped with a smooth, radial, log-convex density, and consider regions of prescribed volume. Brakke's conjecture. Balls centered at the origin are isoperi…

Let $n\geqslant 2$, and let $M^n(c)$ be an $n$-dimensional model space of constant sectional curvature $c\in\mathbb{R}$. Let $K\subset M^n(c)$ be a $\lambda$-convex body, and let…

Let $\Omega$ be a simply connected domain representing the cross-section of a beam, with prescribed area. Its torsional rigidity $T(\Omega)$ is defined by the associated Rayleigh q…

Let $(\overline{X},\bar{g})$ be a smooth compact Riemannian manifold with boundary, and let $\Theta_n(\overline{X},[\bar{g}])$ denote the supremum of the isoperimetric ratio over s…

Let $d\geq 2$ and consider i.i.d. bond percolation on $\mathbb{Z}^d$ with parameter $p>p_c(d)$. Let $\mathcal{C}_\infty$ be the almost surely unique infinite open cluster, let…

Let $\a0\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. Write $0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots$ for its Neumann eigenvalues, let…

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubbl…

Let $(M,g)$ be an $n$-dimensional Cartan-Hadamard manifold with sectional curvature bounded above by $-\kappa^2$ for some $\kappa\geq0$. Let $N^n_\kappa$ be the corresponding model…

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let…

Let $K\subset\mathbb{R}^n$ be a convex body, and let $\Lambda_{\mathrm{FK}}(K)$ denote its normalized minimal first Dirichlet eigenvalue. Restrict to centrally symmetric convex bod…

Let $K\subseteq \mathbb R^n$ be an origin-symmetric convex body. An origin-symmetric convex body $L\subseteq \mathsf{A}K$ is an $O(1)$-perturbation of $\mathsf{A}K$ when its volume…

Let $X$ be a Hadamard space, and let its asymptotic rank be the largest dimension of a Euclidean flat that occurs asymptotically. An isoperimetric filling inequality is a bound on…

Borisenko's conjecture. Among all such sets, the sets maximizing surface area are precisely the lenses of volume $V$.

Let $\Omega\subset\mathbb{R}^2$ be a bounded, simply-connected $C^\infty$-smooth domain, let $b>0$ be the intensity of a homogeneous magnetic field, and let $\mu_1^b(\Omega)>0$ den…

Wulff's conjecture. Minimizers of $\operatorname{Per}_K$ up to null sets are homothetic copies of $K$, and consequently

A hyperbolic $k$-gonal tile is a regular polygon with angles of 120^. Cox's conjecture. A regular $k$-gonal tile with 120^ angles is isoperimetric, meaning that it has least pe…

Higher-dimensional isoperimetric conjecture.

For a polyhedron $P$, define its isoperimetric quotient by … where $V$ is its volume and $F$ its surface area. Two polyhedra are isomorphic when they have the same combinatorial ty…

Let $(M^n,g)$ be a Cartan–Hadamard space, meaning a complete simply connected Riemannian $n$-manifold with non-positive sectional curvature. For a smooth open subset…

Hadamard-type conjecture. (i) $\mathcal G$ is concave if and only if $G_v\geq 0$ for every $x\in\Omega$. (ii) $v$ is $-\infty$-concave, meaning that $\mathcal G$ has convex level s…

Let $Omega$ be an inclusion of fixed volume in a medium and let its electrical polarization tensor be $M$. The matrix $M$ measures the leading dipolar perturbation caused by a unif…

Optimal-scale conjecture. For each $C>0$ there is an $\varepsilon>0$ such that, for every monotone ${\cal F}$, there exists

Isoperimetric-structure conjecture. Given $C$, there are $K,\delta>0$ such that every $(C\log(1/p),p)$-optimal ${\cal F}$ has both the following properties: there is $R\subseteq X$…