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In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.
Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $ge 1$, which contains $X_i$ as convex subsets such that their points are $frac pi2
Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
We show that, in the sense of Baire category, most Alexandrov surfaces with curvature bounded below by $kappa$ have no conical points. We use this result to prove that at most points of such surfaces, the lower and the upper Gaussian curvatures are equal to $kappa$ and $infty$ respectively.
Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large vol