13 problems

Homeomorphism extension. The homeomorphism version of Theorem would also hold for any thin point $x$ with $f_*={\rm id}$ and for any $x\in X_0^2$ if the ball…

Let $X$ be an Alexandrov space, meaning a finite-dimensional complete length space with curvature bounded below in the Alexandrov sense, and suppose that $X$ contains no proper ext…

Conjecture on convex level sets in Alexandrov spaces. The level set $\{f=c\}$ is an Alexandrov space of curvature at least $\kappa$.

Splitting conjecture. If $\phi$ is integrable, then

Weak-integrability conjectures. (1) If $\partial Y=\emptyset$ and $f$ is weakly integrable, then $f$ is integrable. (2) If $\Sigma\in \operatorname{Alex}^n(1)$,…

Fiber-bundle conjecture. If all points in $S$ are weakly $m$-strained, then $\phi:X\to S$ is a fiber bundle map.

Soul conjecture. If $X$ contains an open subset where the curvature is bounded below by a positive constant, then $S$ is a point.

Alexandrov globalization conjecture. Then $(X,d_X)\in\operatorname{Alex}^n(\kappa)$.

Alexandrov gluing conjecture. The gluing produces an Alexandrov space $Y$ if and only if the gluing is by isometry and, for every $p\in Y$, $T_p(Y)$ is a metric cone over…

Sectional-curvature characterization conjecture. The following conditions are equivalent: $({\rm M},{\sf d})$ is an $n$-dimensional Alexandrov space of curvature bounded from below…

Let $X\in\operatorname{Alex}^n(\kappa)$ have no boundary, and let a convex hypersurface in $X$ be equipped with its intrinsic metric. Gromov's conjecture. The hypersurface is an Al…

Let $A\in \operatorname{Alex}^n(0)$ be a noncompact Alexandrov space with a soul $S$. Let $C_i$ be the subsets associated with the soul as in 1.1, and let…

Let $X,Y\in \text{Alex}^n(\kappa)$ be Alexandrov spaces with non-empty boundary, and let $f:\partial X\to\partial Y$ be a homeomorphism. If the gluing $X\cup_fY$ is an Alexandrov s…