On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n2-1/2s) complexity. This answers one of the main open problems from the author’s previous paper [DCG 29, 375-393 (2003)], which provided a weaker upper bound for a restricted class of curves only (graphs of degree-s polynomials). When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n3/2 bound for parabolas.