## Abstract

A point x in a normed space X is said to be singular for a given tiling of X whenever each neighborhood of x intersects infinitely many tiles. We show that, when X is infinite-dimensional and all tiles are convex, special points in the boundary of tiles (like extreme points or PC points, if any) must be singular. Under the further assumptions that X is separable and doesn’t contain c_{0}, singular points abound among the smooth points of any bounded tile. Finally, in any normed space a tiling is constructed which is free of singular points and whose members are both bounded and star-shaped; this disproves the conjecture that Corson’s theorem might apply to star-shaped bounded coverings.

Original language | English |
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Pages (from-to) | 857-868 |

Number of pages | 12 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2000 |

## Keywords

- Convex tile
- Singular point
- Tiling of normed space

## ASJC Scopus subject areas

- Mathematics (all)