Aleksandrov-Fenchel's inequality and intrinsic volumes (2016)
"We study in this thesis the Brunn-Minkowski inequality in the euclidean space and the Aleksandrov-Fenchel inequality for convex bodies. We do this in order to get a better comprehension of the intrinsic volumes (euclidean and spherical) and their properties. We get as a consequence from the Brunn-Minkowski inequality the isoperimetric inequality for convex bodies. Subsequently, we prove the Aleksandrov-Fenchel inequality using mixed volumes and the reproduction of Aleksandrov's proof found in which corresponds to the first proof of the inequality. We talk later about euclidean intrinsic volumes. We see there that the sequence of intrinsic volumes for any convex body is log-concave. Whether the spherical intrinsic volumes are log-concave remains unknown. Our main contribution was to find explicit formulas for the intrinsic volumes of a spherical polygon. Also we proved that this particular sequence of intrinsic volumes is log-concave using the isoperimetric inequality on the sphere."