# On the Dimension of the Singular Set of Perimeter Minimizers in Spaces with a Two-Sided Bound on the Ricci Curvature

## TL;DR

This paper determines the maximum size of irregular points in optimal shapes within certain curved spaces.

## Contribution

The paper proves a sharp upper bound on the dimension of singular sets in perimeter minimizers under Ricci curvature constraints.

## Key findings

- The Hausdorff dimension of the singular set is at most n-5.
- The result applies to noncollapsed limits of manifolds with two-sided Ricci curvature bounds.
- The estimate is shown to be sharp, meaning it cannot be improved.

## Abstract

We show that the Hausdorff dimension of the singular set of perimeter minimizers in noncollapsed limits of manifolds with two-sided bounds on the Ricci curvature is at most \documentclass[12pt]{minimal}
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\begin{document}$$n-5$$\end{document}n-5, where n is the dimension of the ambient space. The estimate is sharp.