Valentine Convex Sets
Here’s a problem which I was introduced to in 1997 in Halifax, failed miserably to solve (along with some other students), and then mentioned at a bar and got many people very confused last week.
You need to know the definition of a convex set: a set is convex if for each two points u, v in the set, the line segment uv also lies in the set.
Define a set to be valentine convex if for every three points u, v, w in the set, at least one of the three line segments uv, vw, uw also lies in the set.
Warm-up: show that if S, T are convex sets, the set (S union T) is valentine convex.
Real problem: is every valentine convex set the union of two convex sets?
Along with this nerdy post I’ll give a shout-out to Prof. Small who was the one to torture us poor high school kids with the problem.
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