## Abstract

We consider the following basic geometric problem: Given 𝜖 ∈ (0, 1/2), a 2-dimensional black-and-white figure is ∊- far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊- far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(𝜖 ^{-4/3} ) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the Ω(𝜖 ^{−3∕2} ) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

Original language | English (US) |
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Pages (from-to) | 413-443 |

Number of pages | 31 |

Journal | Random Structures and Algorithms |

Volume | 54 |

Issue number | 3 |

DOIs | |

State | Published - May 2019 |

## All Science Journal Classification (ASJC) codes

- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics