Sets in a euclidean space which are not a countable union of convex subsets |
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Authors: | M Kojman M A Perles S Shelah |
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Institution: | 1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
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Abstract: | We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds:
(a) |
There is a perfect subsetP⊆S such that for every pair of distinct pointsx, yεP, the convex closure ofx, y is not contained inS.
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(b) (a) |
does not hold and there is a perfect subsetP⊆S such that for every pair of pointsx, yεP the convex closure of {x, y} is contained inS, but for every triple of distinct pointsx, y, zεP the convex closure of {x, y, z} is not contained inS.
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We show that an analogous theorem is impossible for dimension greater than 2. We give an example of a compact planar set with
countable degree of visual independence which is not a countable union of convex subsets, and give a combinatorial criterion
for a closed set inR
d
not to be a countable union of convex sets. We also prove a conjecture of G. Kalai, namely, that a closed planar set with
the property that each of its visually independent subsets has at most one accumulation point, is a countable union of convex
sets. We also give examples of sets which possess a (small) finite degree of visual independence which are not a countable
union of convex subsets. |
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Keywords: | |
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