On a volume estimate for some subsets of a Grassmannian

Measure theory gives us a technique for proving existence theorems. In simple terms this technique can be described as follows: start with a set and put a measure on it. Then find the exact measure of a specified set A. By showing the set of elements B in A that do not satisfy a certain condition P has measure strictly smaller than the measure of the ambient set A, one can deduce the existence of elements in A that satisfies the desired property P. This line of argument can only work when we have methods to find the measure of the set A and also estimate from above the measure of B. This thesis is about one such argument introduced by Ben-Artzi et al. and utilized later successfully by Foias and Olson to prove the existence of a Mane's projection whose inverse is Hölder continuous. The key estimate in those works was an inequality from integral geometry due to Santalo for measures of some subsets of the Grassmannian Space. The aim of this thesis is to expose and improve an alternate argument given by Friz and Robinson where they avoid working on the Grassmannian space and introduce the measures and estimates on a larger space while keeping track of the sizes of the coefficients.