Covering the Crosspolytope with Crosspolytopes

2023/04/30 by Antal Joós, Joós, Antal
Mathematics · #Point processes and geometric inequalities #Limits and Structures in Graph Theory #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2305.00569

Abstract

Let γ^d_m(K) be the smallest positive number λ such that the convex body K can be covered by m translates of λK. Let K^d be the d-dimensional crosspolytope. It will be proved that γ^d_m(K^d)=1 for 1≤ m< 2d, d≥4; γ^d_m(K^d)=(d-1)/(d) for m=2d,2d+1,2d+2, d≥4; γ^d_m(K^d)=(d-1)/(d) for m= 2d+3, d=4,5; γ^d_m(K^d)=(2d-3)/(2d-1) for m= 2d+4, d=4 and γ^d_m(K^d)≤(2d-3)/(2d-1) for m= 2d+4, d≥5. Moreover the Hadwiger's covering conjecture is verified for the d-dimensional crosspolytope.

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