18.434 - Final Project Concentration of Measure

Our understanding of geometry relies heavily on the ability we have to draw graphs and reason about them. This is why we have a good intuitive understanding of 2 and 3−dimensional spaces. This is not the case for highdimensional geometry. The observations we make from low-dimensional cases can not be extrapolated to higher dimensions. Concentration of measure, the study of the distribution of measure in high-dimensional bodies, tries to fill the gap between our intuition and reality. In this paper, I will show basic results of surface and volume distribution in common bodies like spheres and cubes. The punchline for all of these results will be the same: most of the measure of a high-dimensional body is located near any hyperplane through the origin. In particular, for the case of the unit sphere I will show a connection to one of the isoperimetric inequalities. Finally, I will show some applications of the phenomenon of concentration of measure to probability theory and algorithm design. The first application will show that, under certain special mappings from the unit sphere to R, the mapping is concentrated around the median. In a way, this result is reminiscent of the central limit theorem. The second application relates to the proof of the Johnson-Lindenstrauss Lemma, a key result in modern algorithm design. I will not prove this result, but I will sketch how it is related to the main topic and comment on some of its applications.