Computing the Hausdorff boundary measure of semi-algebraic sets

TL;DR

This paper introduces a numerical scheme to accurately approximate moments of the Hausdorff measure on the boundary of semi-algebraic sets, enabling precise computation of boundary-related geometric quantities.

Contribution

It presents a novel numerical method for approximating boundary measure moments of semi-algebraic sets with arbitrary precision.

Findings

Accurately approximates moments of Hausdorff measure on boundaries.

Enables precise computation of boundary length, surface, and integrals.

Provides bounds from above and below for these quantities.

Abstract

Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting quantities like length, surface, or more general integrals on the boundary, as closely as desired from above and below.