6 problems

De Philippis–Gigli's conjecture. The interior $X\setminus\partial X$ is strongly convex: every geodesic joining two points of $X\setminus\partial X$ avoids $\partial X$.

Polyhedral-complex conjecture. If geodesic triangles in orthant spaces $\mathcal{F}_\Omega$ are always closed, then each geodesic polytope $T$ is a polyhedral complex with cells…

Non-closedness conjecture. The convex envelope of three points is not closed, except in very special cases, with the question arising typically in $\mathbb{CP}^2$.

Let $M$ be a Riemannian manifold with the curvature setting of the conjecture, let $\{x_i\}_{i=1}^N\subset B(o,\rho)$, and consider the map $x\mapsto\exp_x(-t\nabla f_2(x))$. Let…

Let $M$ be a Riemannian manifold of nonnegative variable curvature, let $\{x_i\}_{i=1}^N\subset M$, and define tethering as the property that a map preserves every relevant strongl…

Let $M$ be a Riemannian manifold and let $\{x_i\}_{i=1}^N\subset M$. A map $\Psi:M\to M$ is tethered to $\{x_i\}_{i=1}^N$ if, for every strongly convex regular geodesic ball $B$ co…