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Saint Raymond [Studia Math.\\ 80, 63--75, 1984] showed that the limit $$ \\lim_{n\\to\\infty} n^{1/2 + 1/p} \\big(\\text{Vol}\\, B_p^n(\\mathbb{R})\\big)^{1/n^2} $$ exists in $(0,\\infty)$ and provided both lower and upper bounds. 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