Discrete and Computational Geometry最新文献_第2页

$$varepsilon $$-Isometric Dimension Reduction for Incompressible Subsets of $$ell _p$$ $$varepsilon $$的不可压缩子集的等距降维 $$ell _p$$

Discrete and Computational Geometry Pub Date : 2023-10-21 DOI: 10.1007/s00454-023-00587-w

Alexandros Eskenazis

{"title":"$$varepsilon $$-Isometric Dimension Reduction for Incompressible Subsets of $$ell _p$$","authors":"Alexandros Eskenazis","doi":"10.1007/s00454-023-00587-w","DOIUrl":"https://doi.org/10.1007/s00454-023-00587-w","url":null,"abstract":"Abstract Fix $$pin [1,infty )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , $$Kin (0,infty )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and a probability measure $$mu $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>μ</mml:mi> </mml:math> . We prove that for every $$nin mathbb {N}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , $$varepsilon in (0,1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and $$x_1,ldots ,x_nin L_p(mu )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with $$big Vert max _{iin {1,ldots ,n}} |x_i| big Vert _{L_p(mu )} le K$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msub> <mml:mo>max</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> , there exist $$dle frac{32e^2 (2K)^{2p}log n}{varepsilon ^2}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>32</mml:mn> <mml:msup> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:msup> <mml:mi>ε</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> </mml:mrow> </mml:math> and vectors","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Matroids of Gain Signed Graphs 增益带符号图的拟阵

Discrete and Computational Geometry Pub Date : 2023-10-21 DOI: 10.1007/s00454-023-00568-z

Laura Anderson, Ting Su, Thomas Zaslavsky

引用次数: 0

Inscribable Order Types 不可写的订单类型

Discrete and Computational Geometry Pub Date : 2023-10-18 DOI: 10.1007/s00454-023-00591-0

Michael Gene Dobbins, Seunghun Lee

引用次数: 0

An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements 超平面排列特征多项式系数的一个恒等式

Discrete and Computational Geometry Pub Date : 2023-10-18 DOI: 10.1007/s00454-023-00577-y

Zakhar Kabluchko

{"title":"An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements","authors":"Zakhar Kabluchko","doi":"10.1007/s00454-023-00577-y","DOIUrl":"https://doi.org/10.1007/s00454-023-00577-y","url":null,"abstract":"Abstract Consider a finite collection of affine hyperplanes in $$mathbb R^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . The hyperplanes dissect $$mathbb R^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> into finitely many polyhedral chambers. For a point $$xin mathbb R^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> and a chamber P the metric projection of x onto P is the unique point $$yin P$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> minimizing the Euclidean distance to x . The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by $$text {dim}(x,P)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtext>dim</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We prove that for every given $$kin {0,ldots , d}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , the number of chambers P for which $$text {dim}(x,P) = k$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtext>dim</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> does not depend on the choice of x , with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k -th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138 (8), 2873–2887 (2010)].","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"9 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Adjacency Graphs of Polyhedral Surfaces 多面体曲面的邻接图

Discrete and Computational Geometry Pub Date : 2023-10-18 DOI: 10.1007/s00454-023-00537-6

Elena Arseneva, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, Alexander Wolff

{"title":"Adjacency Graphs of Polyhedral Surfaces","authors":"Elena Arseneva, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, Alexander Wolff","doi":"10.1007/s00454-023-00537-6","DOIUrl":"https://doi.org/10.1007/s00454-023-00537-6","url":null,"abstract":"Abstract We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $${mathbb {R}}^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $$K_5$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:math> , $$K_{5,81}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>81</mml:mn> </mml:mrow> </mml:msub> </mml:math> , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $$K_{4,4}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:math> , and $$K_{3,5}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msub> </mml:math> can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (Isr. J. Math. 46 (1–2), 127–144 (1983)), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n -vertex graphs is in $$Omega (nlog n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . From the non-realizability of $$K_{5,81}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>81</mml:mn> </mml:mrow> </mml:msub> </mml:math> , we obtain that any realizable n -vertex graph has $${mathcal {O}}(n^{9/5})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>9</mml:mn> <mml:mo>/</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Evasive Sets, Covering by Subspaces, and Point-Hyperplane Incidences 回避集、子空间覆盖与点-超平面关联

Discrete and Computational Geometry Pub Date : 2023-10-17 DOI: 10.1007/s00454-023-00601-1

Benny Sudakov, István Tomon

{"title":"Evasive Sets, Covering by Subspaces, and Point-Hyperplane Incidences","authors":"Benny Sudakov, István Tomon","doi":"10.1007/s00454-023-00601-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00601-1","url":null,"abstract":"Abstract Given positive integers $$kle d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> and a finite field $$mathbb {F}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> , a set $$Ssubset mathbb {F}^{d}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> is ( k , c )- subspace evasive if every k -dimensional affine subspace contains at most c elements of S . By a simple averaging argument, the maximum size of a ( k , c )-subspace evasive set is at most $$c |mathbb {F}|^{d-k}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mi>F</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> </mml:math> . When k and d are fixed, and c is sufficiently large, the matching lower bound $$Omega (|mathbb {F}|^{d-k})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>F</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of ( k , c )-evasive sets in case d is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of k -dimensional linear hyperplanes needed to cover the grid $$[n]^{d}subset mathbb {R}^{d}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> is $$Omega _{d}big (n^{frac{d(d-k)}{d-1}}big )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which matches the upper bound proved by Balko et al., and settles a problem proposed by Brass et al. Furthermore, we improve the best known lower b","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"266 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135943941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Pizza and 2-Structures 披萨和2-Structures

Discrete and Computational Geometry Pub Date : 2023-10-17 DOI: 10.1007/s00454-023-00600-2

Richard Ehrenborg, Sophie Morel, Margaret Readdy

引用次数: 2

Embeddings of k-Complexes into 2k-Manifolds k-配合物在2k流形中的嵌入

Discrete and Computational Geometry Pub Date : 2023-10-16 DOI: 10.1007/s00454-023-00595-w

Pavel Paták, Martin Tancer

引用次数: 8

Lower Bound on Translative Covering Density of Tetrahedra 四面体平移覆盖密度的下界

Discrete and Computational Geometry Pub Date : 2023-10-16 DOI: 10.1007/s00454-023-00602-0

Yiming Li, Miao Fu, Yuqin Zhang

引用次数: 1

Tropical Compactification via Ganter’s Algorithm 基于甘特算法的热带紧化

Discrete and Computational Geometry Pub Date : 2023-10-14 DOI: 10.1007/s00454-023-00581-2

Lars Kastner, Kristin Shaw, Anna-Lena Winz

引用次数: 0