no. 6
Analyse fonctionnelle
Convex maps on ℝ n and positive definite matrices
Bourin, Jean-Christophe1  ; Shao, Jingjing2
1 Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
2 College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China
Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 645-649.

We obtain several convexity statements involving positive definite matrices. In particular, if A,B,X,Y are invertible matrices and A,B are positive, we show that the map

(s,t)↦TrlogX * A s X+Y * B t Y

is jointly convex on ℝ 2 . This is related to some exotic matrix Hölder inequalities such as

sinh∑ i=1 m A i B i ≤sinh∑ i=1 m A i p 1/p sinh∑ i=1 m B i q 1/q

for all positive matrices A i ,B i , such that A i B i =B i A i , conjugate exponents p,q and unitarily invariant norms ∥·∥. Our approach to obtain these results consists in studying the behaviour of some functionals along the geodesics of the Riemanian manifold of positive definite matrices.

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DOI : 10.5802/crmath.25
Classification : 47A30, 15A60
Bourin, Jean-Christophe 1 ; Shao, Jingjing 2

1 Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
2 College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China 
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     title = {Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices},
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Bourin, Jean-Christophe; Shao, Jingjing. Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices. Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 645-649. doi : 10.5802/crmath.25. http://www.numdam.org/articles/10.5802/crmath.25/

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