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: (nr 1/3 ) for a general n-element matroid with rank r , and (mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was (n log r ) for uniform

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... matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log * n) for uniform matroids were also known.