April 3, 2024Open Access

Random approximation of convex bodies in Hausdorff metric

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JPJoscha ProchnoUniversity of Passau CSCarsten SchüttFachhochschule Kiel MSMathias SonnleitnerJohannes Kepler University of Linz

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Implication

Theoretical analysis reveals exact limiting expected Hausdorff distance in random polygon approximations, indicating asymptotic constant behavior for regular polygons.

Key Points

Expected Hausdorff distance between a polygon and the random convex hull of boundary points exhibits an exact limiting behavior as the sample point count tends to infinity.
Asymptotic constant behavior is derived for a regular polygon from the limiting expected distance, using n independent random points chosen uniformly on the boundary.
Findings clarify random approximation of convex bodies under Hausdorff distance, expanding geometric theory beyond classical intrinsic volumes and symmetric difference metrics.

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Prochno et al. (2024) studied this question.

synapsesocial.com/papers/68e708e7b6db643587683049 https://doi.org/10.48550/arxiv.2404.02870

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Implication

Theoretical analysis reveals exact limiting expected Hausdorff distance in random polygon approximations, indicating asymptotic constant behavior for regular polygons.

Random approximation of convex bodies in Hausdorff metric

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Implication

Key Points

Cite This Study

Random approximation of convex bodies in Hausdorff metric

Discussion

Member takes

Implication

Key Points

Cite This Study

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