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Register a new userFinite Quotient of Join in Alexandrov Geometry
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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 $frac pi2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $frac pi2$ apart and $dim(X_1)+dim(X_2)=n-1$.
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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.
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.
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.
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 volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,mathcal{H}^n)$ has finite topological type.
The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,ldots,K_n)$ of convex bodies $K_1,ldots,K_n$ in $mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,ldots,K_n)$ and $V(K_2,K_2,K_3,ldots,K_n)$. As a consequence, for integers $alpha_1,ldots,alpha_minmathbb{N}$ with $alpha_1+cdots+alpha_m=n$ the product $V_n(K_1)^{frac{alpha_1}{n}}cdots V_n(K_m)^{frac{alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,ldots,m$, is a lower bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$, where $alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$ in terms of the product of the intrinsic volumes $V_{alpha_i}(K_i)$, for $i=1,ldots,m$. The case where $m=2$, $alpha_1=1$, $alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $alpha_1=alpha_2=1$, $alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $alpha_2=cdots=alpha_m=1$, $K_1$ is the unit ball and $K_2,ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.
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