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Journal articles on the topic 'Alexandrov Theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 June 2024

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1

Hijazi, Oussama, Sebastián Montiel, and Simon Raulot. "An Alexandrov theorem in Minkowski spacetime." Asian Journal of Mathematics 23, no. 6 (2019): 933–52. http://dx.doi.org/10.4310/ajm.2019.v23.n6.a3.

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2

Mashiko, Yukihiro. "A splitting theorem for Alexandrov spaces." Pacific Journal of Mathematics 204, no. 2 (June 1, 2002): 445–58. http://dx.doi.org/10.2140/pjm.2002.204.445.

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3

Shen, Zhongmin. "A Regularity Theorem for Alexandrov Spaces." Mathematische Nachrichten 164, no. 1 (1993): 91–102. http://dx.doi.org/10.1002/mana.19931640108.

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4

Lang, Urs, and Viktor Schroeder. "On Toponogov's comparison theorem for Alexandrov spaces." L’Enseignement Mathématique 59, no. 3 (2013): 325–36. http://dx.doi.org/10.4171/lem/59-3-6.

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5

Alexander, Stephanie, and Richard Bishop. "A cone splitting theorem for Alexandrov spaces." Pacific Journal of Mathematics 218, no. 1 (January 1, 2005): 1–15. http://dx.doi.org/10.2140/pjm.2005.218.1.

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6

Chen, Wen-Haw, and Jyh-Yang Wu. "A rigidity theorem for discrete groups." Bulletin of the Australian Mathematical Society 73, no. 1 (February 2006): 1–8. http://dx.doi.org/10.1017/s0004972700038569.

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This work considers the discrete subgroups of group of isometries of an Alexandrov space with a lower curvature bound. By developing the notion of Hausdorff distance in these groups, a rigidity theorem for the close discrete groups was proved.

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7

Su, Xiaole, Hongwei Sun, and Yusheng Wang. "Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650049. http://dx.doi.org/10.1142/s0219199716500498.

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In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

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8

Kuwae, Kazuhiro, and Takashi Shioya. "A topological splitting theorem for weighted Alexandrov spaces." Tohoku Mathematical Journal 63, no. 1 (2011): 59–76. http://dx.doi.org/10.2748/tmj/1303219936.

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9

Wörner, Andreas. "A splitting theorem for nonnegatively curved Alexandrov spaces." Geometry & Topology 16, no. 4 (December 31, 2012): 2391–426. http://dx.doi.org/10.2140/gt.2012.16.2391.

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10

Hua, Bobo. "Generalized Liouville theorem in nonnegatively curved Alexandrov spaces." Chinese Annals of Mathematics, Series B 30, no. 2 (February 18, 2009): 111–28. http://dx.doi.org/10.1007/s11401-008-0376-3.

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11

Danielli, D., N. Garofalo, D. M. Nhieu, and F. Tournier. "The Theorem of Busemann-Feller-Alexandrov in Carnot Groups." Communications in Analysis and Geometry 12, no. 4 (2004): 853–86. http://dx.doi.org/10.4310/cag.2004.v12.n4.a5.

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12

Azagra, Daniel, Anthony Cappello, and Piotr Hajłasz. "A geometric approach to second-order differentiability of convex functions." Proceedings of the American Mathematical Society, Series B 10, no. 33 (October 30, 2023): 382–97. http://dx.doi.org/10.1090/bproc/190.

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We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by C 1 , 1 C^{1,1} convex functions and convex bodies.

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13

Yokota, Takumi. "A rigidity theorem in Alexandrov spaces with lower curvature bound." Mathematische Annalen 353, no. 2 (June 17, 2011): 305–31. http://dx.doi.org/10.1007/s00208-011-0686-8.

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14

Lajara, S., and A. J. Pallars. "A nonlinear map for midpoint locally uniformly rotund renorming." Bulletin of the Australian Mathematical Society 72, no. 1 (August 2005): 39–44. http://dx.doi.org/10.1017/s0004972700034857.

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We provide a criterion for midpoint locally uniformly rotund renormability of normed spaces involving the class of σ-slicely continuous maps, recently introduced by Moltó, Orihuela, Troyanski and Valdiva in 2003. As a consequence of this result, we obtain a theorem of G. Alexandrov concerning the three space problem for midpoint locally uniformly rotund renormings of Banach spaces.

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15

Deville, Robert, and El Mahjoub El Haddad. "The subdifferential of the sum of two functions in Banach spaces II. Second order case." Bulletin of the Australian Mathematical Society 51, no. 2 (April 1995): 235–48. http://dx.doi.org/10.1017/s0004972700014076.

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We prove a formula for the second order subdifferential of the sum of two lower semi continuous functions in finite dimensions. This formula yields an Alexandrov type theorem for continuous functions. We derive from this uniqueness results of viscosity solutions of second order Hamilton-Jacobi equations and singlevaluedness of the associated Hamilton-Jacobi operators. We also provide conterexamples in infinite dimensional Hilbert spaces.

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16

Li, YanYan. "Symmetry of hypersurfaces and the Hopf Lemma." Mathematics in Engineering 5, no. 5 (2023): 1–9. http://dx.doi.org/10.3934/mine.2023084.

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<abstract><p>A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.</p></abstract>

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17

Cai, Qingsong. "An almost isometric sphere theorem and weak strainers on Alexandrov spaces." Indiana University Mathematics Journal 66, no. 4 (2017): 1267–86. http://dx.doi.org/10.1512/iumj.2017.66.6073.

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18

Nagano, Koichi. "A volume convergence theorem for Alexandrov spaces with curvature bounded above." Mathematische Zeitschrift 241, no. 1 (September 2002): 127–63. http://dx.doi.org/10.1007/s002090100409.

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19

Peters, James Francis, and Tane Vergili. "Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems." Applied General Topology 24, no. 1 (April 5, 2023): 25–45. http://dx.doi.org/10.4995/agt.2023.17046.

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This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. In this work, a path cycle is a sequence of maps h1,...,hi,...,hn-1 mod n in which hi : [ 0,1 ] → X and hi(1) = hi+1(0) provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles.

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20

Hua, Bobo, Jürgen Jost, and Shiping Liu. "Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 700 (January 1, 2015): 1–36. http://dx.doi.org/10.1515/crelle-2013-0015.

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AbstractWe apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. Finally, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.

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21

Ge, Jian. "Soul theorem for 4-dimensional topologically regular open nonnegatively curved Alexandrov spaces." Proceedings of the American Mathematical Society 139, no. 12 (December 1, 2011): 4435–43. http://dx.doi.org/10.1090/s0002-9939-2011-10831-1.

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22

Jiao, Zhenhua, and Qiang Li. "The Liouville’s theorem of harmonic functions on Alexandrov spaces with nonnegative Ricci curvature." Indian Journal of Pure and Applied Mathematics 46, no. 1 (February 2015): 51–58. http://dx.doi.org/10.1007/s13226-015-0107-x.

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23

Cao, Jianguo, Bo Dai, and Jiaqiang Mei. "A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces." Frontiers of Mathematics in China 6, no. 1 (August 24, 2010): 35–48. http://dx.doi.org/10.1007/s11464-010-0079-4.

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24

Julin, Vesa, and Joonas Niinikoski. "Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow." Analysis & PDE 16, no. 3 (May 25, 2023): 679–710. http://dx.doi.org/10.2140/apde.2023.16.679.

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25

Haslhofer, Robert, Or Hershkovits, and Brian White. "Moving plane method for varifolds and applications." American Journal of Mathematics 145, no. 4 (August 2023): 1051–76. http://dx.doi.org/10.1353/ajm.2023.a902954.

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abstract: In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.

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26

Mitsuishi, Ayato. "A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications." Geometriae Dedicata 144, no. 1 (June 13, 2009): 101–14. http://dx.doi.org/10.1007/s10711-009-9390-1.

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27

Su, Xiaole, Hongwei Sun, and Yusheng Wang. "A new proof of almost isometry theorem in Alexandrov geometry with curvature bounded below." Asian Journal of Mathematics 17, no. 4 (2013): 715–28. http://dx.doi.org/10.4310/ajm.2013.v17.n4.a9.

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28

Delgadino, Matias G., Francesco Maggi, Cornelia Mihaila, and Robin Neumayer. "Bubbling with L2-Almost Constant Mean Curvature and an Alexandrov-Type Theorem for Crystals." Archive for Rational Mechanics and Analysis 230, no. 3 (July 13, 2018): 1131–77. http://dx.doi.org/10.1007/s00205-018-1267-8.

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29

Gálvez, José A., and Pablo Mira. "A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov." Mathematische Annalen 366, no. 3-4 (December 31, 2015): 909–28. http://dx.doi.org/10.1007/s00208-015-1351-4.

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30

De Gennaro, Daniele, Andrea Kubin, and Anna Kubin. "Asymptotic of the discrete volume-preserving fractional mean curvature flow via a nonlocal quantitative Alexandrov theorem." Nonlinear Analysis 228 (March 2023): 113200. http://dx.doi.org/10.1016/j.na.2022.113200.

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31

Alesker, Semyon. "Valuations on convex functions and convex sets and Monge–Ampère operators." Advances in Geometry 19, no. 3 (July 26, 2019): 313–22. http://dx.doi.org/10.1515/advgeom-2018-0031.

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Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

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32

Desmonts, Christophe. "Spinorial proofs of the Alexandrov Theorem for higher order mean curvatures in Rn+1 and the Heintze–Karcher Inequality." Differential Geometry and its Applications 37 (December 2014): 44–53. http://dx.doi.org/10.1016/j.difgeo.2014.09.003.

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33

Chodosh, Otis, Vishesh Jain, Michael Lindsey, Lyuboslav Panchev, and Yanir A. Rubinstein. "On discontinuity of planar optimal transport maps." Journal of Topology and Analysis 07, no. 02 (March 26, 2015): 239–60. http://dx.doi.org/10.1142/s1793525315500089.

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Consider two bounded domains Ω and Λ in ℝ2, and two sufficiently regular probability measures μ and ν supported on them. By Brenier's theorem, there exists a unique transportation map T satisfying T#μ = ν and minimizing the quadratic cost ∫ℝn ∣T(x) - x∣2 dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, if Λ is convex, T is continuous. We study the reverse problem, namely, when is T discontinuous if Λ fails to be convex? We prove a result guaranteeing the discontinuity of T in terms of the geometries of Λ and Ω in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of ∂Λ to distinguish between Brenier and Alexandrov weak solutions of the Monge–Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

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34

Li, Nan. "Globalization with probabilistic convexity." Journal of Topology and Analysis 07, no. 04 (September 22, 2015): 719–35. http://dx.doi.org/10.1142/s1793525315500223.

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35

Elbert, Maria Fernanda, and Ricardo Sa Earp. "Constructions of $$H_r$$ H r -hypersurfaces, barriers and Alexandrov theorem in $$\mathrm{I\!H}^n\times \mathrm{I\!R}$$ I H n × I R." Annali di Matematica Pura ed Applicata (1923 -) 194, no. 6 (September 13, 2014): 1809–34. http://dx.doi.org/10.1007/s10231-014-0446-y.

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36

Khurana, Surjit Singh. "Vector Measures on Topological Spaces." gmj 14, no. 4 (December 2007): 687–98. http://dx.doi.org/10.1515/gmj.2007.687.

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Abstract Let 𝑋 be a completely regular Hausdorff space, 𝐸 a quasi-complete locally convex space, 𝐶(𝑋) (resp. 𝐶𝑏(𝑋)) the space of all (resp. all, bounded), scalar-valued continuous functions on 𝑋, and 𝐵(𝑋) and 𝐵0(𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀𝑡(𝑋,𝐸), 𝑀 τ (𝑋,𝐸), 𝑀 σ (𝑋,𝐸) of tight, τ-smooth, σ-smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶𝑏(𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.

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37

Delgadino, Matias Gonzalo, and Francesco Maggi. "Alexandrov’s theorem revisited." Analysis & PDE 12, no. 6 (February 7, 2019): 1613–42. http://dx.doi.org/10.2140/apde.2019.12.1613.

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38

WU, Jyh-Yang. "Topological regularity theorems for Alexandrov spaces." Journal of the Mathematical Society of Japan 49, no. 4 (October 1997): 741–57. http://dx.doi.org/10.2969/jmsj/04940741.

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39

Scheepers, Marion. "Finite powers of strong measure zero sets." Journal of Symbolic Logic 64, no. 3 (September 1999): 1295–306. http://dx.doi.org/10.2307/2586631.

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AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

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40

Wang, Mu-Tao, Ye-Kai Wang, and Xiangwen Zhang. "Minkowski formulae and Alexandrov theorems in spacetime." Journal of Differential Geometry 105, no. 2 (February 2017): 249–90. http://dx.doi.org/10.4310/jdg/1486522815.

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41

Panchapagesan, T. V. "Generalization of a Theorem of Alexandroff." Journal of Mathematical Analysis and Applications 189, no. 2 (January 1995): 343–50. http://dx.doi.org/10.1006/jmaa.1995.1022.

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42

Khurana, Surjit Singh. "A Vector Form of Alexandrov's Theorem." Mathematische Nachrichten 135, no. 1 (1988): 73–77. http://dx.doi.org/10.1002/mana.19881350107.

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43

Zhang, Hui-Chun, and Xi-Ping Zhu. "Ricci curvature on Alexandrov spaces and rigidity theorems." Communications in Analysis and Geometry 18, no. 3 (2010): 503–53. http://dx.doi.org/10.4310/cag.2010.v18.n3.a4.

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44

Ene. "HAKE-ALEXANDROFF-LOOMAN TYPE THEOREMS." Real Analysis Exchange 23, no. 2 (1997): 491. http://dx.doi.org/10.2307/44153977.

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45

Dranishnikov, Alexander N., and Dušan Repovš. "On Alexandroff theorem for general Abelian groups." Topology and its Applications 111, no. 3 (April 2001): 343–53. http://dx.doi.org/10.1016/s0166-8641(99)00219-9.

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46

Alexandrov, Victor. "Minkowski-type and Alexandrov-Type Theorems for Polyhedral Herissons." Geometriae Dedicata 107, no. 1 (August 2004): 169–86. http://dx.doi.org/10.1007/s10711-004-4090-3.

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47

Pambuccian, Victor. "Alexandrov–Zeeman type theorems expressed in terms of definability." Aequationes mathematicae 74, no. 3 (December 2007): 249–61. http://dx.doi.org/10.1007/s00010-007-2885-7.

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48

Al-Badry, Ayed E. Hashoosh. "PRE-ALEXANDROFF SPACE." University of Thi-Qar Journal of Science 3, no. 4 (June 28, 2013): 150–55. http://dx.doi.org/10.32792/utq/utjsci/v3i4.532.

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In this paper we introduce a new definition of the topological space weaker of Alexandroff space, namely pre-Alexandroff space. These spaces are which arbitrary intersection of an open set is a pre-open set. In addition to give a new definition of minimal pre-open sets and investigate about some of its properties, also we get some theorems and result related to this space.

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49

Gavriluţ, Alina. "Continuity properties and Alexandroff theorem in Vietoris topology." Fuzzy Sets and Systems 194 (May 2012): 76–89. http://dx.doi.org/10.1016/j.fss.2011.12.010.

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50

Magnanini, Rolando, and Giorgio Poggesi. "On the stability for Alexandrov’s Soap Bubble theorem." Journal d'Analyse Mathématique 139, no. 1 (October 2019): 179–205. http://dx.doi.org/10.1007/s11854-019-0058-y.

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