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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\\). 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