# Best Power in a Probability Inequality

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

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