Abstract
It is well known that isoperimetric regions in a smooth compact (n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1)$$\\end{document}-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.
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