623 problems

Let $C$ be the $\mathbf{\ell}_1$ ball, let $p(w\mid\xi)$ be the posterior density under the uniform prior on $C$, and write $\operatorname{Cov}_{\mathrm{Uni}(C)}(w)$ for the covari…

Let $X$ be a complex normed space and let $n$ satisfy $2\le n<\dim_{\mathbb C}X$. If every pair of $n$-dimensional complex subspaces of $X$ is isometric as metric spaces, then ther…

For integers $d\ge 2$ and $n\ge d+2$, let $F_1,\ldots,F_n\subseteq\mathbb{R}^d$ be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for e…

Improved eta-expanded theorem. The thesis of Theorem …

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$ with the Euclidean metric. For a measurable set $U\subset [0,1]^n$ of locally finite perimeter, let $\operatorname{Vol}(U)$ denote…

Power-polynomial conjecture. Then $\partial K$ is an ellipsoid; consequently, one can take $m=1$ when $n$ is odd and $m=2$ when $n$ is even.

Quadraticity conjecture. Every smooth polynomially integrable hypersurface is a strictly convex quadric in $\mathbb R^{2k+1}$.

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(B_{2m}))$ be the convex hull of the signed Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope…

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(S_{2m}))$ be the convex hull of the Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope…

Böröczky's conjecture. There is a point $x\in P$ with norm

Fradelizi--Meyer's conjecture. For every convex body $K\subset\mathbb{R}^n$,

Let $1\le k\le d$. Let $X$ be a finite transitive subset of the unit sphere in $\mathbb{C}^d$. Unrestricted cylindrical width conjecture. There is a complex $k$-dimensional subspac…

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,…

Three-dimensional flatness conjecture. No hollow convex $3$-body has width larger than $$ . That is,

Let $K=\{p_1,p_2,\dots,p_n\}$ be a finite set in a normed plane $\mathbb{M}^2$, and let $\lambda\geq\lambda_K$. A minimal-arc representation conjecture asserts that there exist bal…

In the setting of Theorem, concerning robustness classes of convex solids and platonic solids, let downward external robustness and downward full robustness be the corresponding do…

Makeev's conjecture. There exists an $A\in SO(n)$ such that all $n(n+1)/2$ of these $(n-1)$-planes are concurrent. The conjecture is a reformulation of Makeev's universal-cover con…

Indefinite energy conjecture. The quantity $Q(\operatorname{\stackrel{\longrightarrow}{Vol}}N)$ is uniquely minimized when $N$ is the intersection of $Q$ with an $a$-plane in $V$ c…

Energy conjecture. The quantity $Q(\operatorname{\stackrel{\longrightarrow}{Vol}}K^+)$ is uniquely minimized when $K$ is an ellipsoid.

Bottleneck conjecture. For convex bodies $K$ in $n$ dimensions with $n$ fixed, the volume $D(K)$ is uniquely minimized when $K$ is an ellipsoid.

Let $K,E\in\mathcal{K}^n$ be convex bodies, and let … be their relative Steiner polynomial. Let $a_1\leq\dots\leq a_n$ be the real parts of the roots of $f(K,E,s)$, and let…

Let $n$ congruent regular unit hexagons form an edge-to-edge connected system. Hexagon polyomino area bound. The area of its convex hull is at most … The statement is presented in…

Let $P$ be a $d$-dimensional polyomino with parameters $l_1,\dots,l_d$ and $v_1,\dots,v_d$, and suppose that the convex hull of $P$ has maximum volume. Let $P'$ be a subpolyomino o…

Let $\mathcal{L}$ be the fixed line segment, $\mathcal{S}$ the square, and let $\mathcal{T}$ denote the triangular objects in the configuration. Consider configurations with…

Let $K\subseteq\mathbb{R}^n$ be centrally symmetric, let $K^\circ$ be its polar body, and let $(\vec{x},\vec{y})$ be uniformly distributed on $K\times K^\circ$. Polar second-moment…