8 problems

Consider the potentials $t^{2k}+5alpha C_{2k}^2(t)$ and $|t|^p$ on the sphere, and let $\mu_{\alpha \to 0}$ denote a minimizer of the first potential in the limit $\alpha \to 0$. L…

Let $S^{n-1}$ be the unit sphere, let $p>0$ be not even, and let ${\operatorname{PFP}}(\mu,p)$ denote the probabilistic $p$-frame potential for a probability measure $\mu$ on…

In the two-input case, let the $p$-frame energy be the energy under consideration, and let $p\notin 2\mathbb{N}$. A minimizer is a measure attaining the minimum of this energy. BGM…

Let $d>2$ in the real case, let $k$ be an integer, and consider configurations optimizing the $p$-frame energy for every $p$ in an interval $[2k-2,2k]$. Finiteness conjecture. Ther…

Let $d\geq 2$ and let $p>0$ with $p\notin 2\mathbb N$. For a probability measure $\mu$ on the sphere, write … for its $p$-frame energy. Discrete-minimizer conjecture. In all dimens…

Let $N=m+kd$ points be given in $\mathbb{S}^{d-1}$, with $1\leq m<d$ and $d\geq 2$, and let $A\in\mathbb{R}^{N\times N}$ be their Gram matrix. For $j=1,\ldots,N$, write…

Universal optimality conjecture for lifted ETFs. The following configurations minimize the $p$-frame potential $\operatorname{FP}_{p,d+1,d}$: when $p\in(p_{k-1},p_k]$, the $L_k^d$…

Let $\Phi^{\alpha,\beta,\delta,\eta}$ be the frame potential considered in the paper, and let Example denote the specified frame or Gram matrix. An absolute-minimizer conjecture as…