We obtain several convexity statements involving positive definite matrices. In particular, if are invertible matrices and are positive, we show that the map
is jointly convex on . This is related to some exotic matrix Hölder inequalities such as
for all positive matrices , such that , conjugate exponents 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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Jean-Christophe Bourin 1; Jingjing Shao 2
@article{CRMATH_2020__358_6_645_0, author = {Jean-Christophe Bourin and Jingjing Shao}, title = {Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices}, journal = {Comptes Rendus. Math\'ematique}, pages = {645--649}, publisher = {Acad\'emie des sciences, Paris}, volume = {358}, number = {6}, year = {2020}, doi = {10.5802/crmath.25}, language = {en}, }
TY - JOUR AU - Jean-Christophe Bourin AU - Jingjing Shao TI - Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices JO - Comptes Rendus. Mathématique PY - 2020 SP - 645 EP - 649 VL - 358 IS - 6 PB - Académie des sciences, Paris DO - 10.5802/crmath.25 LA - en ID - CRMATH_2020__358_6_645_0 ER -
Jean-Christophe Bourin; Jingjing Shao. Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 645-649. doi : 10.5802/crmath.25. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/
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