Local tests for consistency of support hyperplane data

William C. Karl, Sanjeev R. Kulkarni, George C. Verghese, Alan S. Willsky

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


Support functions and samples of convex bodies in Rn are studied with regard to conditions for their validity or consistency. Necessary and sufficient conditions for a function to be a support function are reviewed in a general setting. An apparently little known classical such result for the planar case due to Rademacher and based on a determinantal inequality is presented and a generalization to arbitrary dimensions is developed. These conditions are global in the sense that they involve values of the support function at widely separated points. The corresponding discrete problem of determining the validity of a set of samples of a support function is treated. Conditions similar to the continuous inequality results are given for the consistency of a set of discrete support observations. These conditions are in terms of a series of local inequality tests involving only neighboring support samples. Our results serve to generalize existing planar conditions to arbitrary dimensions by providing a generalization of the notion of nearest neighbor for plane vectors which utilizes a simple positive cone condition on the respective support sample normals.

Original languageEnglish (US)
Pages (from-to)249-267
Number of pages19
JournalJournal of Mathematical Imaging and Vision
Issue number2-3
StatePublished - 1996

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modeling and Simulation
  • Condensed Matter Physics
  • Computer Vision and Pattern Recognition
  • Geometry and Topology
  • Applied Mathematics


  • Computational geometry
  • Data consistency criteria
  • Set reconstruction
  • Support hyperplane data


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