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

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Stephan Lehmich, Patrizio Neff, Johannes Lankeit

Mathematical Physics · Submitted 25 Sep 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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+.

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arXiv ID1209.5535 (version 1)
Pith ID6PLAOBAG, Pith's permanent record of this paper and its review
CitationJohannes Lankeit, Patrizio Neff, Stephan Lehmich. "On the convexity of the function C --> f(det C) on positive definite matrices." arXiv:1209.5535 (2012). https://pith.science/paper/6PLAOBAG
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,394 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.74 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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+. result 0.95 0.95 0.96 asserted 1.00
s1 This generalizes the observation that C --> -ln det C is convex as a function of C. interpretation 0.80 0.24 0.05 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

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