Mahler Conjecture And Related Problems

This researches of this thesis belongs to the theory of convex geometric analysis, anddevoted to the study of Mahler conjecture and related problems. The Mahler volume, firstintroduced by K. Mahler in1939, describes the volume of a convex body times the volumeof its dual body. The maximum of the Mahler volume is provided by the famous Blaschke-Santal′o inequality. A major open problem, which became known as Mahler’s conjectureor as the reverse Blaschke-Santalo′inequality, given by K. Mahler in1939, can be statedas follows:(1) For the centrally symmetric convex bodies, cubes have the minimal Mahlervolumes;(2) For the non-symmetric convex bodies, simplexes have the minimal Mahlervolumes. For over70years Mahler conjecture has attracted the study and attention ofmany mathematicians. For n=2, Mahler himself proved the conjecture. For n≥3,Mahler conjecture is still open. For some special classes of convex bodies, e.g., zonoids,unconditional convex bodies, centred polytopes having at most2n+2vertices, local versionof Mahler conjecture, etc. Bourgain and Milman proved that Mahler conjecture is valid,up to a factor cn, where c is a constant. Many meaning results have been obtained aboutfunctional Blaschke-Santal′o inequality and its reverse form.Our main results can be stated as follows:(1) Using “the vertex removal method”, we give a new proof of the Mahler conjecturein R2.(2) For the case of n=3, we prove that among origin-symmetric bodies of revolution,cylinders have the minimal Mahler volume. Further, we prove that among parallel sectionshomothety bodies,3cubes have the minimal Mahler volume.(3) We give a diferent approach to prove a functional version of the Blaschke-Santal′oinequality due to Ball. Specially, a new approach of defining Steiner symmetrization ofcoercive convex functions is proposed and some fundamental properties of the new Steinersymmetrization are proved. We do not use geometric Steiner symmetrizations and ourapproach is more suitable for certain functional problems.(4) We prove a generalized version of the functional Blaschke-Santal′o inequality dueto Artstein, Klartag and Milman.