Affirmative Resolution of Bourgain’s Slicing Problem Using Guan’s Bound
Bo’az Klartag, Joseph Lehec
Abstract
Abstract We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> such that $$ Vol_{n-1}(K \cap H) > c, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> <mml:mi>o</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> </mml:math> where $c > 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> is a universal constant. Our proof combines Milman’s theory of $M$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> -ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.