16 problems

Let $B_2^d$ and $B_\infty^d$ denote the unit balls of the Euclidean and maximum norms in $P{R}^d$, respectively, and let $\mathrm{vb}(B_2^d,B_\infty^d,n)$ be the corresponding vect…

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…

Square-root discrepancy conjecture. The best bound for this discrepancy, uniformly over all $N$ and all such vector sequences, is of order $\sqrt d$.

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

Let $x_1,\dots,x_k$ be real numbers with $-1<x_i<1$. Assume that, for $i=1,\dots,k$, the cyclic successor is given by … where $x_{k+1}=x_1$. Cyclic doubling permutation conjecture.…

For $n\geq2$, let $\alpha(B_2^n,V)$ and $\beta(B_2^n,V)$ denote respectively the lattice and vector balancing quantities for a convex body $V\in\mathcal{C}_n$, where…

Let $B_2^n$ denote the Euclidean unit ball in $obreakmathbb{R}^n$, let $gamman$ be standard Gaussian measure, and let $beta(U,V)$ be the vector balancing constant of a pair of con…

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 $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ be symmetric matrices whose eigenvalues all lie in $[-1,1]$. Noncommutative vector balancing conjecture. There exist signs…

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 $t$…

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 $u_1,\dots,u_n\in\mathbb{R}^m$ be vectors satisfying $\lVert u_i\rVert_2\leq 1$ for every $i$. A signing is a choice of signs $b_1,\dots,b_n\in\{-1,1\}$. Komlós's 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 $2d$ 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…

Let $X^d$ be a strictly convex $d$-dimensional Minkowski space. The strong-balancing conjecture. … This is proposed as an analogue of the preceding theorem for the quantity measuri…