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Metric Geometry

New submissions

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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06582 [pdf, other]
Title: Contrasting Various Notions of Convergence in Geometric Analysis
Comments: 7 figures by Penelope Chang of Hunter College High School
Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG)

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree.

[2]  arXiv:1803.06610 [pdf, ps, other]
Title: Can You Pave the Plane Nicely with Identical Tiles
Authors: Chuanming Zong
Comments: 10 pages, 16 figures
Subjects: Metric Geometry (math.MG); History and Overview (math.HO)

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Yes, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, three types of hexagons, and fifteen types of pentagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane; there are only two types of octagons and one type of decagons which can form five-fold translative tilings.

[3]  arXiv:1803.06847 [pdf, ps, other]
Title: The square negative correlation on l_p^n balls
Subjects: Metric Geometry (math.MG); Probability (math.PR)

In this paper we prove that for any $p\in[2,\infty)$ the $\ell_p^n$ unit ball, $B_p^n$, satisfies the square negative correlation property with respect to every orthonormal basis, while we show it is not always the case for $1\le p\le 2$. In order to do that we regard $B_p^n$ as the orthogonal projection of $B_p^{n+1}$ onto the hyperplane $e_{n+1}^\perp$. We will also study the orthogonal projection of $B_p^n$ onto the hyperplane orthogonal to the diagonal vector $(1,\dots,1)$. In this case, the property holds for all $p\ge 1$ and $n$ large enough.

Replacements for Tue, 20 Mar 18

[4]  arXiv:1707.04830 (replaced) [pdf, ps, other]
Title: On Banach-Mazur distance between planar convex bodies
Subjects: Metric Geometry (math.MG)
[5]  arXiv:1710.05506 (replaced) [pdf, ps, other]
Title: Multiple Lattice Tilings in Euclidean Spaces
Comments: 6 pages, 2 figures
Subjects: Metric Geometry (math.MG)
[6]  arXiv:1711.02514 (replaced) [pdf, ps, other]
Title: Multiple Translative Tilings in Euclidean Spaces
Comments: 12 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1712.01122, arXiv:1710.05506
Subjects: Metric Geometry (math.MG)
[7]  arXiv:1712.01122 (replaced) [pdf, ps, other]
Title: Characterization of the Two-Dimensional Five-Fold Lattice Tiles
Authors: Chuanming Zong
Comments: 20 pages, 14 figures. arXiv admin note: text overlap with arXiv:1711.02514, arXiv:1710.05506
Subjects: Metric Geometry (math.MG)
[8]  arXiv:1712.09732 (replaced) [pdf, ps, other]
Title: Characterization of the Two-Dimensional Five-Fold Translative Tiles
Comments: 18 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1711.02514, arXiv:1712.01122
Subjects: Metric Geometry (math.MG)
[9]  arXiv:1803.03606 (replaced) [pdf, ps, other]
Title: A Simple proof of Johnson-Lindenstrauss extension
Authors: Manor Mendel
Comments: 2 pages. Elimination of typos
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
[10]  arXiv:1801.00335 (replaced) [pdf, ps, other]
Title: Plato's cave and differential forms
Authors: Fedor Manin
Comments: 33 pages, 1 figure; comments welcome! This version corrects an error pointed out by A. Berdnikov
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Differential Geometry (math.DG); Metric Geometry (math.MG)
[ total of 10 entries: 1-10 ]
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