10 problems

Consider optimal quantization in three-dimensional Euclidean space, and let $V$ denote the polytope whose scaled copies asymptotically describe the interior Voronoi cells in an opt…

Gersho's conjecture. The identity $q_{p,d}=\tilde q_{p,d}$ holds for every $p\geq 1$ and $d\in\mathbb{N}_1$. These constants govern the high-resolution asymptotics of ordinary and…

Let $x$ be a real number and define the probability vector $(p_1,p_2,p_3,\ldots)$ by … For each positive integer $n$, let $\alpha_n$ denote the proposed set of $n$-means, and let…

Let $m\geq 3$, let $x$ be a real number, and define the probability vector $(p_1,p_2,\ldots,p_m)$ by … For each $n$, let $\alpha_n$ denote the proposed set of $n$-means, and let…

Let $P$ be the probability distribution under consideration, let $\alpha^n_{\theta^*,\mu}$ denote the optimally $L^r$-dilated sequence, and let $L^s$-optimality be measured through…

Let $\mathcal{P}$ be the relevant space of probability measures equipped with the convex order, and let a best uniform $r$-approximation mean a best approximation using uniform wei…

Sharp Gaussian maximal-radius conjecture. The liminf bound is sharp:

Let $d_W(N)$ denote the critical dimension used in the optimal quadratic quantization of one-dimensional Brownian motion, and let $N$ be the quantization level. Logarithmic growth…

Exponential dilatation conjecture. The sequence $(\alpha_n^{\theta^{\star},0})_{n\geq 1}$ is asymptotically $L^s$-optimal. Numerical experiments for an exponential distribution sup…

Gaussian dilatation conjecture. The sequence $(\alpha_n^{\theta^{\star},m})_{n\geq 1}$ is asymptotically $L^s(P)$-optimal for every $s>0$. Numerical experiments for the one-dimensi…