have nonempty intersection. We may then consider the collection where we replace the two sets and with the single set . In this new collection, every subcollection of sets will have nonempty intersection. The inductive hypothesis therefore applies, and shows that this new collection has nonempty intersection. This implies the same for the original collection, and completes the proof. Colorful Helly theorem The colorful Helly theorem is an extension of Helly's theorem in which, instead of one collection, there are d+1 collections of convex subsets of . If, for every choice of a transversal – one set from every collection – there is a point in common to all the chosen sets, then for at least one of the collections, there is a point in common to all sets in the collection. Figuratively, one can consider the d+1 collections to be of d+1 different colors. Then the theorem says that, if every choice of one-set-per-color has a non-empty intersection, then there exists a color such that all sets of that color have a non-empty intersection. Fractional Helly theorem For every a > 0 there is some b > 0 such that, if are n convex subsets of , and at least an a-fraction of (d+1)-tuples of the sets have a point in common, then a fraction of at least b of the sets have a point in common. See also Carathéodory's theorem Kirchberger's theorem Shapley–Folkman lemma Krein–Milman theorem Choquet theory Radon's theorem, and its generalization, Tverberg's theorem Notes References . . . Heinrich Guggenheimer (1977) Applicable Geometry, page 137, Krieger, Huntington . . . . Theorems in convex geometry Theorems in discrete geometry Articles containing proofs Geometric transversal theory