A stability criterion to the moment theorem

Abstract

For a domain D, a pointpand a function f the integral M f (D; p)= Z D f(px)dx is called the moment of D with respect to p taken with the function f. Herepx denotes the distance of x to p. The Moment Theorem of László Fejes Tóth states the following: Let H be a convex polygon in E 2 with at most six sides and f a non-increasing function defined for non-negative reals. Let p1;pn be distinct points and let Di be the Dirichlet cell of pi relative to H. Then we have n P i=1 M f (Di; pi) 5 nM f (Hn; o); where Hn is a regular hexagon of area a(Hn)=a(H)=n centered at o. In the paper a stability criterion to the Moment Theorem is established.