Negative Entropy Function and Properties

The negative entropy function in $mathbf{R}^n$ is the function $f : mathbf{R}^n rightarrow mathbf{R}$ , with domain the set of vectors with positive components, and values on the domain given by

$f(x) = -sum_{i=1}^n x_i log x_i .$

This function is convex.

Proof: Since the function is a sum of functions, each of which depends on one variable not appearing in the others, we just need to check the convexity of the function of one variable

f(xi) = left{ begin{array}{ll} -xi log xi & mbox{if } xi > 0, +infty & mbox{otherwise.} end{array} right.

The convexity of the latter derives directly from the second-order condition:

$frac{d^2}{dxi^2} f(xi) = frac{1}{xi} > 0 .$