The m-core properly contains the m-divisible points in space

doi:10.1016/0167-8655(93)90138-4

Pattern Recognition Letters

Volume 14, Issue 9, September 1993, Pages 703-705

https://doi.org/10.1016/0167-8655(93)90138-4 Get rights and content

Abstract

A point x in ®^d is an m-divisible point of a finite set of n points S ⊂ ®^d if x is contained in conv S_b i=1, …, m, for some partition S=S₁OS₂O…OS_m of S. Let D_m (S) denote the set of m-divisible points of S. We say x is the m-core of S if each closed half-space containing x also contains at least m points of S. Let C_m(S) denote the set of points in the m-core. Reay, Sierksma and others have conjectured that conv D_m(S)=C_m(S). In this note we give a counterexample for n=9, d=3, m=3. The conjecture is known to be true for d=2.

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^☆: Research supported by NSERC grant No. A3013, NSERC/JSPS Bilateral Exchange Program and FCAR grant No. EQ 1678.

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The m-core properly contains the m-divisible points in space☆

Abstract

On 3N points in a plane

Primitive Radon partitions for cyclic polytopes

Israel J. Math.

MSc. Project

Two constructional problems in aligned spaces

Zastos. Mat. (Appl. Math.)