k-sets and continuous motion in R3

We prove several new results concerning k-sets of point sets on the 2-sphere (equivalently, for signed point sets in the plane) and k-sets in 3-space. Specic results include spherical generalizations of (i) Lov asz’ lemma (regarding the number of spherical k-edges intersecting a given great circle) and of (ii) the crossing identity for kedges due to Andrzejak et al. As a new ingredient compared to the planar case, the latter involves the winding number of k-facets around a given point in 3-space, as introduced by Lee and by Welzl, independently. As a corollary, we obtain a crossing identity for the number of pinched crossings (crossing pairs of triangles sharing one vertex) of k-facets in 3-space.

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