Abstract:We study \(\Gamma_{2^n}(K)\), the least positive number \(\gamma>0\) such that an \(n\)-dimensional convex body \(K\) can be covered by \(2^n\) translates of \(\gamma K\). For \(n\)-simplices \(\Delta_n\), we prove that \(\Gamma_{2^n}(\Delta_n)\), as a sequence in \(n\), tends to \(1/2\). For the cross-polytope \(B_1^n\), we show that \(\Gamma_{2^n}(B_1^n)\leq5/6\) holds for all \(n\geq2\), and that \(\limsup_{n\to\infty}\Gamma_{2^n}(B_1^n)\leq0.641\cdots\). Finally, we prove the existence of a constant \(\kappa_*<1\) such that \(\Gamma_{2^n}(B_p^n)\leq\kappa_*\) for all \(n\geq2\) and all \(p\in[1,\infty]\).
From: Senlin Wu [view email]
[v1]
Sun, 31 May 2026 07:58:54 UTC (20 KB)
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