Let $B_p^n$ be the unit ball in the standard $p$-th norm in $R^n$. Suppose we intersect this ball with a codimension one hyperplane $H$ and ask the following question: for which $H$ is the volume of this section maximal? Ball showed that $H$ perpendicular to $(1,1,0,\ldots,0)$ gives the maximal section for the cube (case $p=+\infty$). Later Meyer and Pajor found maximal sections for $1<p<2$, whereas the case $p>2$ remained open. We shall show that Ball's direction gives the maximizer for $p>10^{15}$, covering a part of the problematic regime. Based on a joint work with Tomasz Tkocz and Alexandros Eskenazis.
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