24 problems

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…

Non-approximability conjecture. There exist Alexandrov spaces with curvature bounded below that admit no approximation by a sequence of Riemannian manifolds of fixed dimension with…

Let $X$ be an Alexandrov space and let $E\subset X$ be a primitive extremal subset, meaning an extremal subset that is not obtained as a proper intersection of larger extremal subs…

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.

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space with an upper curvature bound in the sense of Alexandrov, namely a $\operatorname{CAT}(\kappa)$ space. Topol…

Boundary curvature conjecture. It is still conjectured that $\partial X$ itself is an Alexandrov space with the same lower curvature bound $\kappa$ as $X$.

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

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 a two-dimensional $\operatorname{CBB}(\kappa)$ space contain $\lambda$-convex domains homeomorphic to a disk, and fix their area. Area-constrained inradius conjecture. Among al…

Higher-dimensional curvature conjecture. In higher dimension, $\tilde S_c$ is also an Alexandrov space with curvature $\geq0$ and an empty boundary.

An Alexandrov space is a metric space with curvature bounded below in the Alexandrov sense, and the closure of the space of Riemannian manifolds with a lower curvature bound refers…

Conjecture 0.3. If $|X_1X_2|\geq\frac\pi2$ and $n_1+n_2=n-2$, then either the $X_i$ belong to a compact convex subset of dimension $n-1$ in $X$, or $X$ is isometric to a spherical…

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…

Primitive extremal subset conjecture. Every primitive extremal subset of an Alexandrov space, equipped with its induced intrinsic metric, should be an Alexandrov space with the sam…

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 $Y$ be an $m$-dimensional Alexandrov space with curvature bounded below by $-1$, and let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below by $-1$. Suppos…

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…

Let $Z$ be a compact metric space of Hausdorff dimension $n$, let $X$ be a metric space, and let $f:Z\to X$ be a distance non-increasing onto map. Assume that $X$ and $Z$ are Alexa…

Let a sequence of Alexandrov spaces with curvature bounded below collapse to a limit space. An extremal subset is a distinguished subset of an Alexandrov space preserved by its geo…