ETD

• Bobkov's estimate of moments
• Brunn-Minkowski inequality
• Fomin derivative
• Gaussian measure
• Krugova's dichotomy
• log-concave measure
• s-concave measure
• Skorohod derivative

Log-concave measures arise naturally in the context of probabilistic optimization, as pointed out by Prékopa, and serve the role of a reference measure in infinite dimensional vector spaces for purposes such as gradient flows. They have been thoroughly studied first by Borell, then by Brascamp and Lieb. In this thesis we present the basic theory regarding concave measures (characterization in the finite dimensional case, stability under projection and disintegration, integrability of seminorms, zero-one law), then we present some results such as estimates of the moments and a dichotomy property, proved by Krugova, regarding the differentiability.