Nina Lebedeva
We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.
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Uniform boundedness on extremal subsets in Alexandrov spaces
本文研究了满足特定维数、曲率和直径条件的Alexandrov空间，证明了其极值子集的数量、Betti数和体积均一致有界，核心方法是应用Yamaguchi引入的本质覆盖技术。
We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal subset has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regul…
Application of good coverings to collapsing Alexandrov spaces
本文针对无真极小子集的低维坍缩目标Alexandrov空间，改进Perelman结果，利用Mitsuishi-Yamaguchi好覆盖构造三元组的同伦正合列与上同调谱序列，并将结果推广至本原极小子集。
Alexandrov spaces are CS sets
本文证明了Alexandrov空间的极值分层是Siebenmann意义下的CS分层，并证明了无真极值子集的Alexandrov空间方向空间同胚于球面，多面体情形下该结论对迭代方向空间同样成立。
The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov's original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extre…
This paper is devoted to prove that if an Alexandrov space of curvature not less than $κ$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space obtained by gluing the extremal subset along the isometry is an Alexandrov space of curvature not less than $κ$. This is a generalization of Perelman's doubling and Petrunin's gluing theorems.
In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrappier-Wang \cite{LeW2010volent} proved that the universal cover $\tilde{M}$ is isometric to the hyperbolic space $\mathbb{H}^n$. We will prove analogue theorems for Alexandrov spaces.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $Z/\sim$, where $z, z'\in\partial Z$ and $z\sim z'$ if only if $f(z)=f(z')$. We will partially verify this conjecture, and give a classification for compact Alexandrov $n$-spaces with relatively maximal volume. We will also give an elementary proof for a pointed version of Bishop-…
In this paper, we introduce quasi-convex subsets in Alxandrov spaces with lower curvature bound, which include not only all closed convex subsets without boundary but also all extremal subsets. Moreover, we explore several essential properties of such kind of subsets including a generalized Liberman theorem. It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alxandrov spaces with lower curvature bound.
Lipschitz homotopy convergence of Alexandrov spaces II
本文研究了Alexrov空间的Lipschitz同伦收敛，提出了适用于更广泛情形的同伦稳定性定理，并展示了如何保持奇异性strata的同伦构造。
We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $π/2$, where we prove some rigidity for spaces. Although we consider Alexandrov spaces with curvature bounded below, the results are new even in the Riemannian case.
We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below…
Extremal subsets in geodesically complete spaces with curvature bounded above
本文在曲率有上界的测地完备空间（GCBA空间）中引入了极小子集的概念，证明了在附加假设下其拓扑奇点集构成极小子集，并探讨了相关的结构性质。
We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the ellipsoid. Simple examples with non-unique minimal or maximal ellipsoids conclude this article.
We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.
We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.
The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its re ected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume achieves in that case when the hyperplane goes through on a vertex and orthogonal to the height of the simplex at this vertex. More interesting that in the higher dimensional cases this position is not optimal. We also determine the optimal position of hyperplane in the 5-d…
Epimorphism classes and relatively maximal metrics in large-scale geometry
本文研究了大规模几何中满射类与相对极大度量的性质，通过码域度量的相对极大性条件刻画了极端满射。
A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms,…
Extremal affine surface areas in a functional setting
本文在函数空间中引入极值仿射表面积，建立其对偶公式，证明其与函数积分幂成正比，从而推广并改进了凸体情形的相关结果。
Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
本文刻画了光滑射影簇上超临界自由除子类集合的 Lefschetz 型算子核空间，解决了 Shenfeld-van Handel 的公开问题，并给出了 Alexandrov-Fenchel 与 Khovanskii-Teissier 不等式极值的代数几何证明。
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