Is the min of nonnegative functions concave?

Question

Let X,Y be nonnegative functions. I was wondering if $E(\min(X,Y))\leq \min(E(X),E(Y))$? For that it would be useful to have min of nonnegative functions as a concave function to apply Jensen's inequality. Is the min concave?

Any help would be appreciated.

Answer 1

Probabilistic Jensen's inequality does apply to multivariate scalar functions too.

The mapping $x_1,x_2,\ldots,x_n\to \max(x_1,x_2,\ldots,x_n)$ is convex on $\mathbb{R}^n$ (proved directly) which gives $$ \mathbf{E}\max(X_1,\ldots,X_n)\geqslant \max(\mathbf{E}X_1,\ldots,\mathbf{E}X_n). $$ By taking $X_i:=-X_i$ we obtain also $$ \mathbf{E}\min(X_1,\ldots,X_n)\leqslant \min(\mathbf{E}X_1,\ldots,\mathbf{E}X_n). $$