Superresolution of principal semi-algebraic sets

TL;DR

This paper establishes the H"older continuity of the truncated moment map near principal semi-algebraic sets using volume bounds and convex optimization, with applications to potential transforms in two variables.

Contribution

It introduces a novel proof of H"older continuity for the moment map near semi-algebraic sets, combining algebraic geometry and convex optimization techniques.

Findings

H"older continuity of the truncated moment map is proven near principal semi-algebraic sets.

The method applies volume bounds of semi-algebraic sets and convex optimization.

Main estimates are used for potential transforms in two variables, relevant to perturbations of quadrature domains.

Abstract

The H\"older continuity of the truncated moment map of a shade function in Euclidean space is established in the vicinity of a principal semi-algebraic set. The proof combines volume bounds of semi-algebraic sets and convex optimization methods. The main estimate is applied to a potential type transform specific to two real variables, for perturbations of quadrature domains.