Discrete & Computational Geometry

, Volume 53, Issue 3, pp 569–586 | Cite as

On the Reconstruction of Convex Sets from Random Normal Measurements



We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error \(\eta \), we provide upper bounds on the number of probes that one has to perform in order to obtain an \(\eta \)-approximation of this convex set with high probability. Our result relies on the stability theory related to Minkowski’s theorem.


Surface reconstruction Minkowski problem Surface area measure 

Mathematics Subject Classification

52A27 52A39 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, LJKGrenobleFrance
  2. 2.CNRS, LJKGrenobleFrance

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