10 problems

Let $M:\mathbb{R}^{d\times d}\to\boxed{?}$ be a linear map, and let … be the Schatten-1 ball. Volume-ratio conjecture. Is it true that … A positive answer would give an…

K51mlf3s conjecture. There exists a universal constant $K$ such that every square matrix $A$ whose columns have unit $\ell_2$ norm satisfies

Let $K\subseteq\mathbb{R}^d$ be a zonotope. For symmetric convex bodies, the vector balancing constant $\operatorname{vb}(K,K)$ is the least $r\geq0$ such that every finite sequenc…

Let $T$ be a positive integer, let $v_1,\ldots,v_T\in\mathbb{B}_2^d$ be adversarial input vectors, and let $[T]=\{1,\ldots,T\}$. A signing is a vector $x\in\{\pm1\}^T$. Strong Koml…

Let $v_1,\ldots,v_T\in\mathbb{R}^m$ be a sequence of vectors with Euclidean norm at most $1$. In the oblivious adversarial online setting, the algorithm receives $v_t$ at time $t$…

Komlós's conjecture. There exists a constant $K>0$ such that, for every such matrix $A$,

Let $c(d,n,k)$ denote the extremal signed-sum quantity studied in the paper, and let $\omega_{d+1}$ be the corresponding extremal vector configuration. For a linear subspace…

Let $n$ be a positive integer, and let $\bm{v}_1,\dots,\bm{v}_n\in\mathbb{R}^n$ satisfy $\|\bm{v}_i\|_2\leq 1$ for $i\in[n]$. Komlós' conjecture. There exists a constant $C$ such t…

Let $\overline{\mathcal{C}}(k,d)$ denote the extremal quantity for $k$-collapsing families of unit vectors in $d$-dimensional normed spaces. The $2d$ extremal-value conjecture. … T…

Let $\overline{\mathcal{C}}(k,d)$ denote the extremal quantity for $k$-collapsing families of unit vectors in $d$-dimensional normed spaces. The $2d-1$ threshold conjecture. … The…