26 problems

Zaporozhets–Tarasov conjecture. For every convex body $K$ in $\mathbb{R}^d$,

Normalization-exponent conjecture. The normalization exponent $\alpha=q+(k+3)/2$ remains unchanged for such sampling distributions.

Let $a$ denote the fixed parameter from the definition of $A$, let $b>0$, and let $\gamma_n[b]$ denote the Gaussian measure or distribution appearing in $A(a,\gamma_n[b])$. The qua…

Learned-bias RVFL conjecture. RVFL with learned biases outperforms RVFL without learned biases most strongly in the regime of high-dimensional balls with small radii.

Extension conjecture. The conclusion of the nontrivial barycentre theorem should hold for every $p\geq 2$, and likely also for every $p\geq 1$.

Let $d$ and $n$ be as in the beta-polytope model, and fix parameters $\beta_1,\ldots,\beta_n\geq -1$. Write $\mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n}$ for the corresponding beta…

Disk maximality conjecture. There exists some $\delta>0$ such that, for every such $X$ and every $l$ with $0<l<\delta$,

Let $\bigcirc^d$ denote the Euclidean unit ball in dimension $d$, and let $P_{\bigcirc^d}^{(d)}(n)$ be the probability that $n$ independent uniformly random points in this ball are…

Floor-probability bounds conjecture. For all $n\geq 3$ and all $K\in\operatorname{CCSF}([0,1])$,

Lower-dimensional volume conjecture. A $d$-dimensional volume for an $m$-dimensional object, where $d<m$, undergoing a fragmentation process follows the Benford distribution for th…

Rectangle-perimeter conjecture. The perimeters of the sub-rectangles from a rectangle undergoing an unrestricted fragmentation process converge to Benford behavior.

Lower-dimensional volume conjecture. Even lower-dimensional volumes under the unrestricted fragmentation process follow Benford's Law.

Let $n$ points be sampled from a distribution in dimension $d>1$, and let $\mathrm{LF}$ denote the leapfrog distance between points in the resulting sample. Leapfrog-distance conje…

Let $f$ be an admissible probability density function on $\mathbb{R}^d$, with support $S$ having $K$ path-connected components, and let $n$ points be sampled according to $f$. For…

Let $r$ denote the separation parameter for two unit balls in the sum-of-norms clustering model, and let the threshold separating the regime where SON clustering fails to separate…

For a locally finite point set in the region … let the maximal layers be induced by a homogeneous Poisson point process, where $F:[0,\alpha^{1/(d-1)}]^{d-1}\to[0,\infty)$ has conti…

Let $U,W\neq\mathbb C$ be simply connected planar domains containing $0$, let $T^V$ be the first exit time of planar Brownian motion from $V$, and define … A domain $W$ is spiral-l…

Let $E_k$ be the simplex of multinomial probability parameters, let … and, for $R\geq 1/k$, let … For $R\in(1/k,1)$, choose $j$ such that … where $\nu_j=\sqrt{\frac{j}{k(k-j)}}$ is…

Let $\operatorname{Arm}(n,d,w)$ be the space of open $n$-edge arms in $\mathbb{R}^d$ with fixed edgelength $w$, and let $\operatorname{Pol}(n,d,w)$ be the corresponding space of cl…

Roychowdhury's conjecture. The Voronoi regions of the points in an optimal set of two-means partition the disc into two semicircles.

Let $U=(u_1,u_2,\dots)$ be a sequence of points in $[0,1]^m$, with each point $u_i=(x_1,x_2,\dots,x_m)$ chosen independently and uniformly from $[0,1]^m$, and let…

Let $K$ range over the class $\mathcal K_d$ of convex bodies in $\mathbb R^d$, and let $\widehat K_n$ be the convex hull of $n$ independent observations uniformly distributed on…

Let $n$ be the number of directions in the star of needles, let $M$ be the associated maximum intersection-count parameter, and let $p(i,\alpha)$ denote the probability of exactly…

Let $D$ be a bounded domain with bounded principal curvatures. A semi-stable rubber band in $D$ is a minimal rubber band, without requiring it to be strictly minimal. Consider the…

Let $c_n$ denote the best possible upper bound for the volume ratio $V_n(K)/V_n(L)$ among convex bodies whose projections satisfy the covering-shadow condition. In dimension $3$, c…