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Let $f:S^{n-1}\rightarrow \mathbb{R}_{+}$ be a Lipschitz function. For $1\leq k\leq n$, define $f_k:G_{n,k}\rightarrow \mathbb{R}_{+}$ by $f(E)=\max_{x\in S^{n-1}\cap E}f(x)$. Let $\sigma_k$ denote the Haar probability measure on $G_{n,k}$, and let $\sigma = \sigma_1$. For every number $t>0$, one can prove that $\sigma(x\in S^{n-1} s.t. f(x)\geq t)\leq \sigma_k(E\in G_{n,k}:f_k(E)\geq t)$. What is the maximal power $m>0$, such that $\sigma(x\in S^{n-1} s.t. f(x)\geq t)\leq (\sigma_k(E\in G_{n,k}:f_k(E)\geq t))^{m}$?

Thanks, Yuval

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    $\begingroup$ What is the definition of $G_{n, k}$? $\endgroup$ – Daniel Apr 1 at 14:24
  • $\begingroup$ It is the space of k dimensional subspaces of the n’th dimensional Euclidean space. $\endgroup$ – Yuval Apr 3 at 3:51

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