On the convexity of the function C --> f(det C) on positive definite matrices

TL;DR

This paper characterizes the conditions under which the composition of a twice-differentiable function with the determinant is convex on positive definite matrices, generalizing known convexity results for the negative logarithm of the determinant.

Contribution

It provides a necessary and sufficient condition on the function f for the convexity of f composed with the determinant on positive definite matrices.

Findings

Derived a precise inequality involving f'' and f' for convexity.

Generalized the convexity of -ln det C to a broader class of functions.

Established a complete characterization of convexity conditions.

Abstract

We prove a condition on f \in C^2(\R+,\R) for the convexity of (f o det) on PSym(n), namely that f o det is convex on PSym(n) if and only if f"(s)+(n-1)/(ns) f'(s) >= 0 and f'(s)<= 0 \forall s \in \R+. This generalizes the observation that C --> -ln det C is convex as a function of C.