On Extremal Sections of Subspaces of \(L_p\)

Abstract

Let \(m,n\in {\mathbb {N}}\) and \(p\in (0,\infty )\). For a finite dimensional quasi-normed space \(X=({\mathbb {R}}^m, \Vert \cdot \Vert _X)\), let

$$\begin{aligned} B_p^n(X) = \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \Vert x_i\Vert _X^p \leqslant 1\right\} . \end{aligned}$$

We show that for every \(p\in (0,2)\) and X which admits an isometric embedding into \(L_p\), the function

$$\begin{aligned} S^{n-1} \ni \uptheta = (\uptheta _1,\ldots ,\uptheta _n) \longmapsto \left| B_p^n(X) \cap \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \uptheta _i x_i=0 \right\} \right| \end{aligned}$$

is a Schur convex function of \((\uptheta _1^2,\ldots ,\uptheta _n^2)\), where \(|\cdot |\) denotes Lebesgue measure. In particular, it is minimized when \(\uptheta =\big (\frac{1}{\sqrt{n}},\ldots ,\frac{1}{\sqrt{n}}\big )\) and maximized when \(\uptheta =(1,0,\ldots ,0)\). This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body \((B_p^n(X))^\circ \) if the unit ball \(B_X\) of X is in Lewis’ position. Finally, we prove a lower bound for the volume of projections of \(B_\infty ^n(X)\), where \(X=({\mathbb {R}}^m,\Vert \cdot \Vert _X)\) is an arbitrary quasi-normed space.

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Acknowledgements

I would like to thank Franck Barthe, Apostolos Giannopoulos, Olivier Guédon, Assaf Naor and the anonymous referees for constructive feedback on this work. I am also very grateful to Tomasz Tkocz for many helpful discussions.

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Correspondence to Alexandros Eskenazis.

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This work was completed while the author was in residence at the Institute for Pure & Applied Mathematics at UCLA for the long program on Quantitative Linear Algebra. He would like to thank the organizers of the program for the excellent working conditions. He was also supported in part by the Simons Foundation.

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Eskenazis, A. On Extremal Sections of Subspaces of \(L_p\). Discrete Comput Geom 65, 489–509 (2021). https://doi.org/10.1007/s00454-019-00133-7

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Keywords

  • \(\ell _p\)-ball
  • Subspaces of \(L_p\)
  • Extremal sections
  • Lewis’ position
  • Volume
  • Mean width

Mathematics Subject Classification

  • Primary 52A40
  • Secondary 52A20, 52A21