Geometric lower bounds for parametric matroid optimization

We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k-sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: Omega (nr(1/3)) for a general n-element matroid with rank r, and Omega (m alpha(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was n (n log r) for uniform matroids; upper bounds of O (mn(1/2)) for arbitrary matroids and O (mn(1/2)/log* n) for uniform matroids were also known.