Nous développons un point de vue de théorie de l'information sur certains problèmes de géométrie des convexes, fournissant par exemple une nouvelle propriété d'équipartition des mesures de probabilités log-concaves, une inégalité de comparaison gaussienne de l'entropie de mesures log-concaves, une formulation entropique de la conjecture de l'hyperplan, et une nouvelle inégalité inverse concernant l'entropie exponentielle pour des mesures log-concaves, analogue à l'inégalité inverse Brunn–Minkowski due à V. Milman.
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn–Minkowski inequality.
@article{CRMATH_2011__349_3-4_201_0, author = {Sergey Bobkov and Mokshay Madiman}, title = {Dimensional behaviour of entropy and information}, journal = {Comptes Rendus. Math\'ematique}, pages = {201--204}, publisher = {Elsevier}, volume = {349}, number = {3-4}, year = {2011}, doi = {10.1016/j.crma.2011.01.008}, language = {en}, }
Sergey Bobkov; Mokshay Madiman. Dimensional behaviour of entropy and information. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 201-204. doi : 10.1016/j.crma.2011.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.008/
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