Local tests for consistency of support hyperplane data

Abstract

Abstract. Support functions and samples of convex bodies in R ~ 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.

Eliciting beliefs

“…However, in more complicated settings, verification of (9) can be computationally burdensome. Karl et al (1995) demonstrate that condition (9) needs to be verified only for (S + 1)−tuples of vectors y i 1 , . .…”

“…Karl et al (1995) provide necessary and sufficient conditions for consistency of support function measurements. Some new terminology is necessary.…”

Three‐dimensional support function estimation and application for projection magnetic resonance imaging

“…This section provides a customized summary of the theory of support functions for the case of 3D based on the paper by Karl et al (1995). Our emphasis is on providing an intuitive description of properties and their use in the context of projection reconstruction.…”

“…An algorithm for estimating 2D support functions was developed by Prince and Willsky (1990) with implementations reported by Lele et al (1992) for a laser-radar application and by Huff (1997, 1998) in the context of PET. Karl et al (1995) extended the theory to arbitrary dimensions but did not translate their results into an algorithm. To the best of our knowledge, this paper is the first of its kind to address the problem of actually estimating a 3D support function, albeit for a fairly restricted geometry.…”

“…We describe how to obtain projection data from k-space as well as how to set up the imaging equations needed for an algebraic reconstruction. Section III provides an introduction to the theory of support functions based on the paper by Karl et al (1995) together with a description of our algorithm for setting up the tests needed for making an inconsistent support function estimate consistent. We also show how to apply duality theory (Bertsekas and Tsitsiklis, 1989) to turn the associated optimization problem into a simple nonnegative least squares problem.…”