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Cauchy-Schwartz inequality and geometric incidence problems
Dr. Misha Rudnev
The Cauchy-Schwartz inequality is essentially a rather obvious
statement that the cosine of an angle cannot exceed one in the absolute
value. The implications are vast though, and we shall discuss some. For
instance, we can use this inequality to show that for a large number
n of straight lines and n points in the plane (no matter how they
are arranged), the total number of incidences (i.e. pairs (point, line)
such that the point lies on the line) cannot exceed some n^{3/2}
(rather than the obviuos bound n^2). Or that a body of volume V
has at least one shadow whose area exceeds some V^{2/3}.
This page last updated 26th September 2005