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.