Abstract
We prove that a collection of compact convex sets of bounded diameters in R d that is unbounded in k independent directions has a k-flat transversal for k< d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1.
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