# Concentration inequality for Lipschitz functions with orthogonal gradients

Let $$f_j:\mathbb{R}^n\to\mathbb{R}$$ be a set of 1-Lipschitz functions for $$1\leq j\leq M$$. From Gaussian isoperimetry or a log-Sobolev inequality, it can be shown that $$\mathbf{Pr}(|f_j(X)-\mathbf{E}f_j(X)|\geq t) \leq C\exp(-ct^2)$$ where the probability distribution over $$X$$ is the standard Gaussian.

I wonder if there exist conditions which guarantee any extra concentration for the variable $$\min_{1\leq j\leq M} |f_j(X) - \mathbf{E} f_j(X)|.$$ For example, if $$n=M$$ and $$f_j(X)$$ is simply the $$j$$-th coordinate, then one gains a factor of $$M$$ in the exponent.

This makes me wonder if a sufficient condition is to ask that $$\langle f_j,f_k\rangle = 0$$ when $$j\not= k$$. The intuition is that the level sets of the medians of $$f_j$$ are orthogonal to each other, so their intersections have higher codimension, and in particular the volume of their union is roughly additive. Is this correct?