LOCAL TESTS FOR CONSISTENCY OF SUPPORT HYPERPLANE DATA

Support functions and samples of convex bodies in R(n) are studied wit h regard to conditions for their validity or consistency. Necessary an d sufficient conditions for a function to be a support function are re viewed in a general setting. An apparently little known classical such result for the planar case due to Rademacher and based on a determina ntal inequality is presented and a generalization to, arbitrary dimens ions 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 o f samples of a support function is treated. Conditions similar to the continuous inequality results are given for the consistency of a set o f discrete support observations. These conditions are in terms of a se ries of local inequality tests involving only neighboring support samp les. Our results serve to generalize existing planar conditions to arb itrary dimensions by providing a generalization of the notion of neare st neighbor for plane vectors which utilizes a simple positive cone co ndition on the respective support sample normals.