Inequalities for the Gamma function and estimates for the volume of sections of 𝐵ⁿ_{𝑝}

Bastero, Jesús; Galve, Fernando; Peña, Ana; Romance, Miguel

doi:10.1090/S0002-9939-01-06139-1

Inequalities for the Gamma function and estimates for the volume of sections of $B^n_p$
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by Jesús Bastero, Fernando Galve, Ana Peña and Miguel Romance PDF

Proc. Amer. Math. Soc. 130 (2002), 183-192 Request permission

Abstract:

Let $B^n_p=\{(x_i)\in \mathbb {R}^n;\sum _1^n|x_i|^p\leq 1\}$ and let $E$ be a $k$-dimensional subspace of $\mathbb {R}^n$. We prove that $|E\cap B^n_p|_k^{1/k}\geq |B^n_p|_n^{1/n}$, for $1\leq k\leq (n-1)/2$ and $k=n-1$ whenever $1<p<2$. We also consider $0<p<1$ and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

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