8 problems

Let $I=(-1,1)$, and let $\varphi_1^\alpha$ be the ground state eigenfunction for the symmetric stable process of index $0<\alpha<2$ killed upon leaving $I$. Concavity conjecture. T…

Real-rootedness concavity conjecture. If $\bar f_{\overrightarrow{a}}$ is real-rooted, then $f_{\overrightarrow{a}}$ is $\frac1p$-concave on $\Gamma_n$; equivalently,…

Diagonal concavity conjecture. The function $f_{\overrightarrow{a}}$ is $\frac1p$-concave on $\Gamma_n$ if and only if it is $\frac1p$-concave on $\Delta$, if and only if…

Let $p \in [0,1]$. Let $\mu$ be a symmetric measure in $\mathbb R^n$ with an $\alpha$-concave density function, where $\alpha \geq -\frac{p}{n}$. Let $\oplus_p$ denote the $p$-Fire…

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let $\ddot v_s(N)=v_s(N-1)-2v_s(N)+v_s(N+1)$. Let $s^\dagger=15.04807\ldots$, and assume t…

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let ${\cal C}_+(s)$ be the set of integers $N$ at which strict local concavity fails, so t…

Minus-one concavity conjecture. The map $N\mapsto v_{-1}(N)$ is locally strictly concave for all $N>2$, equivalently $\ddot v_{-1}(N)<0$ for all $N>2$. Numerical data suggest this…

Let $b_0,b_1,\ldots,b_k$ be fixed vectors in $\mathbb{R}_{++}^n=(0,\infty)^n$, where $0\leq k<n$. For vectors $v_1,\ldots,v_n$, write $\operatorname{per}(v_1,\ldots,v_n)$ for the p…