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

Abstract

Let $n\ge 2$ . We prove a condition on $f\in C^2({ℝ}_{+},ℝ)$ for the convexity of f ∘det on $\mathbb{P}\mathbb{S}\text{ym}(n)$ , namely that f ∘det is convex on $\mathbb{P}\mathbb{S}\text{ym}(n)$ if and only if

$$f^{″}(s)+\frac{n-1}{ns}\cdot f^{\prime }(s)\ge 0\text{and}{f}^{\prime }(s)\le 0\forall s\in {ℝ}_{+}.$$

This generalizes the observation that C ↦ − ln det C is convex as a function of C.