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Function space topologies defined by sectional integrals and applications to an extremal problem

Published online by Cambridge University Press:  24 October 2008

K. J. Falconer
K. J. Falconer
Affiliation:
Corpus Christi College, Cambridge
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Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 1 , January 1980 , pp. 81 - 96
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

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