An exercise from Aubrun-Szarek book: Alice and Bob Meet Banach The Interface of Asymptotice Geometric Analysis and Quantum Information Theory:
I want to know how to prove this: Exercise 5.46 - Let $f$ be a 1-Lipschitz and positive function on (S^{n-1}, g) with $n>1$. Set $q = (Ef^2)^{1/2}$. Show that for any $t > 0$, $P(f \geq q + t) \leq \exp(-nt^2/2)$ and $P(f \leq q - t) \leq e \exp(-nt^2/2)$.