22 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…

Let $\alpha\in[0,1]$, let $\mathcal N_\alpha$ be the scaled Newton polygon … and let $\mathcal H_\alpha(\xi,\eta)$ denote the entropy function obtained from the coefficient asympto…

Lions's conjecture. For a general $f$, the superlevel sets of $u$ should be convex.

Let $\Omega\subset\mathbb{R}^N$ be a suitable domain and let the Robin ground state be the positive first eigenfunction of the Laplace operator on $\Omega$ with positive Robin para…

A symmetric form is a function on tuples of positive-definite matrices that is invariant under simultaneous unitary conjugation; in the paper, the relevant results are the generali…

Let $u_0$ be a bounded nonnegative function in $\mathbf{R}^N$ with compact support. For each $1\leq\alpha<2$, Theorem 3.4 asserts that there exists $T_\alpha>0$ such that…

Let $A\in\mathcal{H}_n^+$, let $B\in\mathbb{C}^{n\times n}$, let $p$ be the parameter appearing in the Epstein concavity theorem, and let $\operatorname{Tr}_k$ denote the $k$-trace…

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 $B_2^n$ denote the unit Euclidean ball in $\mathbb{R}^n$. For a measurable set $A\subset\mathbb{R}^n$, the parallel volume is the function $t\mapsto\mu(A+tB_2^n)$, where $\mu$…

For $y\in(0,1)$, consider the function $p\mapsto\sin_p(y)$, where $\sin_p$ is the generalized sine. Concavity threshold conjecture. There exists $p_0\in(0,1)$ such that…

For each $s$, let ${\cal C}_+(s)$ be the set of $N$-values at which the actual map $N\mapsto v_s(N)$ is strictly convex, equivalently where local strict concavity fails. Defect-set…

Let $v_s^x(N)$ be the empirically computed minimal average standardized Riesz pair-energy for $N$ points on $\mathbb{S}^2$. Empirical concavity-threshold conjecture. There exists…

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 $v_s(N)$ denote the minimal average standardized Riesz pair-energy of $N$ points on the circle $\mathbb{S}^1$, and let local strict concavity mean $v_s(N-1)-2v_s(N)+v_s(N+1)<0$…

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…

Concave-hull growth-rate conjecture.

Concavity conjecture. Concavity is attained only in an appropriate limit with the $(l,r,w,L)$ ensemble.

Let $s$ be a proper local scoring rule, and suppose that $s$ can be generated from some concave gauge choice. The standard gauge choice is $\phi=q_0s$, and write $u_i=q_i/q_0$ for…