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12 pages, 279 KB

Open AccessArticle

Construction of ε-Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric

by Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu

Mathematics 2025, 13(8), 1358; https://doi.org/10.3390/math13081358 - 21 Apr 2025

Viewed by 512

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et [...] Read more.

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing

ε

-nets for

(K^{n}, d_{B M})

based on finite subsets of

Z^{n}

theoretically. In this work, we present an algorithm to construct

ε

-nets for

(K^{2}, d_{B M})

and a

(1 / 4)

-net for

(C^{2}, d_{B M})

is constructed. To the best of our knowledge, this is the first concrete

ε

-net for

(C^{2}, d_{B M})

for such a small

ε

. Full article

(This article belongs to the Section B: Geometry and Topology)

23 pages, 360 KB

Open AccessArticle

Homothetic Covering of Crosspolytopes

by Yunfang Lyu, Feifei Chen and Senlin Wu

Mathematics 2025, 13(4), 546; https://doi.org/10.3390/math13040546 - 7 Feb 2025

Cited by 1 | Viewed by 696

Abstract

The exact value of

Γ_{m} (K)

, which is the least positive number

γ

such that a convex body K can be covered by m translates of

γ K

, is usually difficult to obtain. We present exact values of [...] Read more.

The exact value of

Γ_{m} (K)

, which is the least positive number

γ

such that a convex body K can be covered by m translates of

γ K

, is usually difficult to obtain. We present exact values of

Γ_{14} (B_{1}^{3})

Γ_{11} (B_{1}^{4})

Γ_{2 n} (B_{1}^{n})

Γ_{2 n + 1} (B_{1}^{n})

, and

Γ_{2 n + 2} (B_{1}^{n})

, where

B_{1}^{n}

is the unit ball of

R^{n}

endowed with the taxicab norm. Full article

(This article belongs to the Section B: Geometry and Topology)

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16 pages, 318 KB

Open AccessArticle

An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies

by Xiangyang Han, Senlin Wu and Longzhen Zhang

Axioms 2024, 13(2), 132; https://doi.org/10.3390/axioms13020132 - 19 Feb 2024

Cited by 3 | Viewed by 1661

Abstract

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. Full article

(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)

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15 pages, 388 KB

Open AccessArticle

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

by Man Yu, Yafang Lv, Yanping Zhao, Chan He and Senlin Wu

Mathematics 2023, 11(9), 2000; https://doi.org/10.3390/math11092000 - 23 Apr 2023

Cited by 5 | Viewed by 1701

Abstract

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented. Full article

(This article belongs to the Section B: Geometry and Topology)

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