Property 𝑃₃ and the union of two convex sets

Buchman, E. O.

doi:10.1090/S0002-9939-1970-0259750-3

Property $P_{3}$ and the union of two convex sets
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by E. O. Buchman

Proc. Amer. Math. Soc. 25 (1970), 642-645

DOI: https://doi.org/10.1090/S0002-9939-1970-0259750-3

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Abstract:

A set $S$ in a linear space is said to have the three-point convexity property ${P_3}$ iff for each triple of points $x,\;y,\;z$ of $S$, at least one of the segments $xy,\;xz,\;yz$ is a subset of $S$. It is proved that if $S$ is a compact set in Euclidean space of dimension at least three with at least one point interior to its convex kernel and if the set of points of local nonconvexity of $S$ is interior to its convex hull, then $S$ has property ${P_3}$ iff it is the union of two convex sets.

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