7 problems

Critical exponent conjecture. The critical value of $b1$ for the rigidity-flexibility dichotomy is one of

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{U}$ and $\mathbb{V}$ be the two functions associated with $f$ in the paper's Bellman-function formulation.…

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{B}_\varepsilon$ denote the minimal function associated with $f$ on the domain $\Omega_\varepsilon$. $C^{1,1}$ regul…

Let $S^1=\{y\in\mathbb{R}^2\mid y_1^2+y_2^2=1\}$, and let $\mathfrak{B}^\circ_\delta[f]$ and $\mathfrak{B}^{\circ,*}_\delta[f]$ denote, respectively, the minimal locally concave an…

The paper studies the multidimensional extension of a Kyle–Back model, where the construction involves a convex potential and the associated Monge–Ampère equation. Monge–Ampère con…

Let $\Omega \subset \mathbb{R}^n$ be a compact convex polytope, and let $\Gamma_k$ denote its $k$-skeleton. A convex function is a function whose epigraph is convex. Higher-dimensi…

Simplex and cube extremality conjecture. (i) Among all bounded open convex sets in $\mathbb R^n$ having a fixed positive volume, the $n$-dimensional regular simplex—the interior of…