13 problems

Xu's uniqueness conjecture. For all $d$, $|T_d(x)|\le 1$, and the points in $\mathcal S_+$ and $\mathcal S_-$ are the only points on the face $x_1+\cdots+x_d=1$ at which…

For $d\ge 3$, let $T_d$ be the recursively defined polynomial in the elementary symmetric functions on the standard simplex $T^d$, and write $\|T_d\|_{C(T^d)}$ for its uniform norm…

Filliman's conjecture. The minimum $k$-dimensional projection volume is attained by $T_{\ast}$:

Let $\hat{m}^{(\mathrm{LL})}_{n,b}$ and $\hat{m}^{(\mathrm{NW})}_{n,b}$ denote the local linear and Nadaraya–Watson estimators with Dirichlet kernel, respectively. Near the boundar…

Let $S_4$ be the real four-dimensional simplex, and let $F_5$ be the set of $126=\binom{9}{4}$ points constructed by placing the specified boundary points on every face and the rem…

Let $S_3$ be the real three-dimensional simplex, and let $F_5$ be the set obtained by taking the boundary points described in the paper together with the four interior points … max…

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_j$ be the associated fundamental Lagrange polynomials.…

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N_4=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_i$ be its fundamental Lagrange polynomials. The degre…

Let $S_d$ be the real $d$-simplex, and let $F_3\subset S_d$ consist of its $d+1$ vertices, the $2\binom{d+1}{2}$ edge points $tV_i+(1-t)V_j$ with $i\ne j$ and $t=(1+1/\sqrt{5})/2$,…

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 $s>0$, let $\mathbb S^{p-1}\subset\mathbb R^p$ be the unit sphere, and consider the unconstrained polarization problem $\mathcal P_s^*(\mathbb S^{p-1},N)$, in which the $N$ poi…

Expected-value error bound. One has

Let $S$ be the simplex, let $E_S$ be its set of edge directions, let $\widetilde d_S(\pmb x,\pmb\xi)$ be the associated coefficient, and let $\widetilde P_S(D)$ be the correspondin…