Abstract
A family S of sets in R d is sundered if for each way of choosing a point from r≤d+1 members of S, the chosen points form the vertex-set of an (r−1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R d, and for each partition (S ′, S ″), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S ′ from the members of S ″. This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.
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Lewis, T., von Hohenbalken, B. & Klee, V. Common supports as fixed points. Geom Dedicata 60, 277–281 (1996). https://doi.org/10.1007/BF00147364
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DOI: https://doi.org/10.1007/BF00147364