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

2012/09/25 by Stephan Lehmich, Lehmich, Stephan, Patrizio Neff +3
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Functional Equations Stability Results #Mathematical Inequalities and Applications #Mathematical Physics (math-ph) #Optimization and Variational Analysis #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1209.5535

arxiv created 2012/09/25 · openalex publication_date 2012/09/25 · arxiv updated 2012/09/26 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We prove a condition on f ∈ C²(\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 ∀ s ∈ \R+. This generalizes the observation that C --> -ln det C is convex as a function of C.

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