Spring 2019

Volumes of Sublevel Sets of Homogeneous Polynomials

Tuesday, Apr. 30, 2019 3:30 pm4:15 pm PDT

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Consider the compact sub-level set $K:=\{x: g(x)\leq1\}$ of a nonnegative homogeneous polynomial $g$. We show that its Lebesgue volume ${\rm vol}(K)$ can be approximated as closely as desired by solving a sequence of generalized eigenvalue problems with respect to a pair of Hankel matrices of increasing size, whose entries are obtained in closed form. The methodology also extends to compact sets of the form $\{x: a\leq g(x)\leq b\}$ for non-homogeneous polynomials with degree $d\ll n$. It reduces the volume computation in $R^n$ to a ``volume" computation in $R^d$ (where $d={\rm deg}(g)$) for a certain pushforward measure. Another extension to computing volumes of finite intersections of such sub-level sets is also briefly described.