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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Updating Hardy, Littlewood and Pólya with Linear Programming
verfasst von : Larry Shepp
Erschienen in: The Mathematics of Preference, Choice and Order
Verlag: Springer Berlin Heidelberg
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Some of the standard inequalities that mathematicians use can be proven with convexity arguments or linear programming.
1
Perhaps others cannot, so we might say that an inequality is “simple” if there is a convexity based proof. The Cauchy-Schwarz inequality, which may be the most famous and useful inequality ever found is simple in this sense Steele (2004), but there are so many proofs of it that it seems that almost any method will give one, so it may be that it is simple in any sense. The Schwarz inequality can be stated for a general measure space but it easily reduces to the statement that
$$EX^2 EY^2 \ge (EXY)^2 $$
where
X
and
Y
are any r.v.'s on a common probability space, Ω.. Equality holds if and only if
X
and
Y
are proportional.