Wednesday, February 29th, 2012
3:00pm - 4:00pm, in Science 2-064

Eric Grinberg

UMass Boston

Comparing Volume by Comparing Area: the Busemann-Petty Problem

Abstract: The classic Cavalieri Principle states that the volumes of two objects are equal if their corresponding parallel cross-sectional areas are equal. (Indeed, this is often presented in a 1st year calculus course.) Hence, if one object has greater parallel cross-sectional areas it also has greater volume. But what if we take cross-sections passing through a fixed point ("concurrent cross-sections")? Can volume comparisons be realized by means of such data? This leads to the Busemann-Petty Problem, first posed in 1956, whose solution is contained in a series of papers culminating in the late 1990s. Extensions and generalizations of this problem have formed a field of considerable current activity, and many related questions remain open. We will discuss the history of the problem, present some motivating examples and counterexamples, and prove in an elementary way one Busemann-Petty type inference. We will also propose problems and directions for future investigation. The talk is intended to be accessible to undergraduate students.