On levels in arrangements of curves. II. A simple inequality and its consequences

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(n^2-12s/)) complexity. This answers one of the main open problems from the author's previous paper, which provided a weaker bound for a restricted class of curves (graphs of degree-s polynomials) only. 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(n³2/) bound for parabolas.