7 problems

Strict descent conjecture. Every convex polygon satisfies the strict descent condition. The condition is introduced to construct electrostatic skeletons where different Schwarz ref…

Let $\mathcal{C}_{N}$ be the class of convex polygons in $\mathbb{R}^2$ with at most $N$ sides and perimeter $2\pi$, and let $\Omega^*$ be a maximizer of the first nonzero Steklov…

Existence and non-degeneracy conjecture. The supremum is attained by a convex polygon $\Omega^*$ with exactly $N$ sides. In particular, the maximizer cannot be a degenerate polygon…

Let $k$ be a constant, let $\alpha$ satisfy $\alpha\leq k-1$, and let $\Delta_k'(n)$ denote the normalized extremal area parameter for convex $k$-gons defined in the paper. The nor…

Let $\Delta_4'(n)$ denote the minimum, over configurations of $n$ points in the normalized setting considered in the paper, of the area of a convex quadrilateral containing four po…

Pre-compactness conjecture. The sequence $\{P^{\ell}\}$ is pre-compact modulo affine transformations. That is, this sequence has a convergent subsequence which converges to another…

Let $\Gamma_n$ denote the set of convex $n$-gons with each edge of length $1$, and let $E_n^\alpha$ be the energy associated with the increasing continuous function…