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The lower dimensional slicing inequality for functions and related distance inequalities (2405.10223v3)

Published 16 May 2024 in math.MG

Abstract: It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb Rn$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb Rn$ of codimension $k$ such that [ \int_{K \cap F} f \geq ck (d_{\rm ovr}(K, \mathcal{BP}kn)){-k} ] where $d{\rm ovr}(K, \mathcal{BP}kn)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a universal constant. The upper bound $d{\rm ovr}(K, \mathcal{BP}kn) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right){3/2}$ was established in [13] for every origin-symmetric convex body $K$. In this note we show that there exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous probability density $f$ supported on $K$ such that for every subspace $F$ of codimension $k$, [ \int{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right){-k}. ] As a consequence we obtain a lower bound for $d_{\rm ovr}(K, \mathcal{BP}kn)$ with $K$ a convex body, complementing the upper bound in \cite{koldobsky2011isomorphic}. This is [c \sqrt{n/k} (\log(n)){-1/2} \leq \sup_K d{\rm ovr}(K, \mathcal{BP}_kn) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right){3/2}.] The case $k=1$ was obtained previously in [5,6].

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References (16)
  1. Estimates for moments of general measures on convex bodies. Proceedings of the American Mathematical Society, 146(11):4879–4888, 2018.
  2. E. D. Gluskin. Extremal properties of orthogonal parallelepipeds and their applications to the geometry of banach spaces. Mathematics of the USSR-Sbornik, 64(1):85, 1989.
  3. W. Gregory and A. Koldobsky. Inequalities for the derivatives of the Radon transform on convex bodies. Israel Journal of Mathematics, 246(1):261–280, 2021.
  4. J. Haddad and A. Koldobsky. Radon transforms with small derivatives and distance inequalities for convex bodies. arXiv preprint arXiv:2312.16923, 2023.
  5. B. Klartag and A. Koldobsky. An example related to the slicing inequality for general measures. Journal of Functional Analysis, 274(7):2089–2112, 2018.
  6. B. Klartag and G. V. Livshyts. The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding, pages 43–63. Springer International Publishing, Cham, 2020.
  7. A. Koldobsky. A functional analytic approach to intersection bodies. Geometric & Functional Analysis GAFA, 10:1507–1526, 2000.
  8. A. Koldobsky. Fourier analysis in convex geometry. Number 116. American Mathematical Soc., 2005.
  9. A. Koldobsky. A hyperplane inequality for measures of convex bodies in ℝn,n≤4superscriptℝ𝑛𝑛4\mathbb{R}^{n},n\leq 4blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_n ≤ 4. Discrete & Computational Geometry, 47(3):538–547, 2012.
  10. A. Koldobsky. A n𝑛\sqrt{n}square-root start_ARG italic_n end_ARG estimate for measures of hyperplane sections of convex bodies. Advances in Mathematics, 254:33–40, 2014.
  11. A. Koldobsky. Slicing inequalities for measures of convex bodies. Advances in Mathematics, 283:473–488, 2015.
  12. Measure comparison and distance inequalities for convex bodies. Indiana University Mathematics Journal, 2019.
  13. Isomorphic properties of intersection bodies. Journal of Functional Analysis, 261(9):2697–2716, 2011.
  14. A. Markoe. Analytic tomography, volume 13. Cambridge University Press, 2006.
  15. E. Milman. Generalized intersection bodies. Journal of Functional Analysis, 240(2):530–567, 2006.
  16. S. J. Szarek. Nets of Grassmann manifold and orthogonal group. In Proceedings of research workshop on Banach space theory (Iowa City, Iowa, 1981), volume 169, page 185. University of Iowa Iowa City, IA, 1982.

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