Orlicz Projection Bodies

As Schneider [50] observes, the classical Brunn-Minkowski theory had its origin at the turn of the 19th into the 20th century, when Minkowski joined a method of combining convex bodies (which became known as Minkowski addition) with that of ordinary volume. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is that of projection body (precise definitions to follow). Four decades ago, in a highly influential paper, Bolker [1] illustrated how Minkowski’s projection operator, its range (called the class of zonoids), and its polar were in fact objects of independent investigation in a number of disciplines. Within the Brunn-Minkowski theory, the two classical inequalities which connect the volume of a convex body with that of its polar projection body are the Petty and Zhang projection inequalities. In retrospect, it is interesting to observe that these inequalities did not emerge out of Blaschke’s school, and that it took seven decades from Minkowski’s discovery of projection bodies, for the Petty projection inequality to appear (see e.g., the books by Gardner [11], Leichweiss [21], Schneider [50], and Thompson [53] for reference). It took yet another two decades for the Zhang projection inequality to be discovered. Establishing the analogs of the Petty and Zhang projection inequalities for the projection operator (as opposed to the polar projection operator) are major open problems within the field of convex geometric analysis. Unlike the classical isoperimetric inequality, the Petty and Zhang projection inequalities are affine isoperimetric inequalities in that they are inequalities between a pair of geometric functionals whose ratio is invariant under affine transformations. The Petty projection inequality is not only stronger than (i.e., directly implies) the classical isoperimetric inequality, but it can be viewed as an optimal isoperimetric inequality. In the same manner that the classical isoperimetric inequality leads to (in fact is equivalent to) the Sobolev inequality, the Petty projection inequality has lead to Zhang’s affine Sobolev inequality [56] that is stronger than (directly implies) the classical Sobolev inequality and yet is independent of any underlying Euclidean structure. In the early 1960’s, Firey (see e.g. Schneider [50]) introduced an Lp-extension of Minkowski’s addition (now known as Firey-Minkowski Lp-addition) of convex bodies. In the mid 1990s, it was shown in [30,31], that when Firey-Minkowski Lpaddition is combined with volume the result is an embryonic Lp-Brunn-Minkowski theory. This theory has expanded rapidly. (See e.g. [2–8,14–19,22–28,30–43,49,51, 52,54,55].) An early achievement of the new Lp Brunn-Minkowski theory was the discovery of Lp-analogs of projection bodies and of the Petty projection inequality [33], with an alternate approach to establishing this inequality presented by Campi and