32 problems

Let $\bar c_L$ be the normalized Lagrangian capacity on $2n$-dimensional ellipsoids, let $P(1/n,\dots,1/n)$ denote the cube with all $n$ radius parameters equal to $1/n$, and let…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form $\omega_{\mathrm{std}}$. A Lagrangian torus…

Let $0<a_1\le a_2\le\cdots\le a_n<\infty$, and let $E^{2n}(a_1,\dots,a_n)\subset\mathbb{C}^n$ be the ellipsoid defined by … For a closed Lagrangian torus $L\subset(X,\omega)$, set…

Classification conjecture. Every closed totally magnetic submanifold of $E(\mathbf a)$ is of the form

Let $K,L\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $L\subseteq K$. Two-body flat-shadow conjecture. If for every $p\in\partial K$ there are at least $L(n)$ point light s…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body with boundary of class $C^3$. Local light-source conjecture. If for every $p\in\partial K$ there are at least $n+1$ point lig…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the in…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a strictly convex body, and let $\mathcal S$ be a family of hyperplanes that is the image of an embedding of the tangent bundle of the sphe…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the in…

For $q\geq 1$, consider the ellipsoid … Ellipsoid existence conjecture. There is a non-planar free boundary minimal immersed disk into this ellipsoid if and only if $q>3$. This is…

Let $S\subset \mathbb{R}^{n}$ be an embedding of $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ that is $O$-star, and let $\phi:S\to S$ map each point to the unique point on the ray from…

Let $0<a_1\leq a_2\leq\dots\leq a_n<\infty$, and let $E^{2n}(a_1,a_2,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form. Ellipsoid extremal-torus conjecture.…

Let $L,K\subset \mathbb{R}^{n}$ be convex bodies, with $n\geq 3$, and suppose that $L\subset \operatorname{int}K$. For $x\in\mathbb{R}^{n}\setminus K$, let $S(L,x)$ denote the boun…

Let $K,L\subset\mathbb R^3$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is strictly convex. For $u,w\in\mathbb S^2$ with $u\bot w$, let $\pi_u$ denote…

Usher's conjecture. Between the center of $A_{0,n}$ and $\mathrm{accum}(L_{n,0})$, the function $c_{L_{n,0}}$ equals

Non-ellipsoid positive John family conjecture. The determinant of the matrix component of this family is not identically constant: there is no constant $C$ such that

Let $1\le p\le\infty$ and let the semiaxes be $\sigma_j=j^{-\lambda}$ for $j\in\mathbb{N}$, with $\lambda>\frac{1}{s}=\left(\frac{1}{2}-\frac{1}{p}\right)_+$. Write…

Let $K,L\subset\mathbb R^n$ be convex bodies with $L\subset\operatorname{int}K$. For each $x\in\partial K$, let $\Sigma(L,x)$ denote the graze of $L$ from $x$, namely the intersect…

Let $n\ge1$, $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$, and $a=(a_1,\dots,a_n)\in(0,+\infty)^n$ be as in the Laguerre-integral conjecture, and let $K_\alpha(a)$ denote the c…

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 $B_1(n)$ denote the unit ball in $\mathbb{R}^n$, let $\mathbb{S}^{n-1}$ denote the unit sphere, and let $L oindent oise$ be a line. For each $x otin K$, let $Cx$ be the cone ci…

Let $B_1(n)$ denote the unit ball in $\mathbb{R}^n$, and let $\mathbb{S}^{n-1}$ denote the unit sphere. For a convex body $K ot i \mathbb{S}^{n-1}$ contained in $B1(n)$, and each…

Let $E(a,b)$ denote the four-dimensional symplectic ellipsoid, let $B^4(c)=E(c,c)$ be the round four-ball, and for $N\in\mathbb{Z}_{\geq 1}$ define … Also let $f_0(x)$ be the corre…

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 $n\geq 2$ and $k\geq 0$. For each $i\geq 0$, let $A_{i,n}^{(k)}$ and $\hat{A}_{i,n}^{(k)}$ be the associated classes, and let $C_{L_{n,k}}(\alpha)$, $L_{n,k}$, $c_{i,n,k}$,…