Abstract
All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complex-valued valuations on spherical Lebesgue measurable functions are also completely classified.
Highlights
IntroductionValuations on convex bodies can be considered as valuations on suitable function spaces
A function z defined on a lattice (L, ) and taking values in an Aabelian semigroup is called a valuation if (1)for all f, g L
Valuations on convex bodies can be considered as valuations on suitable function spaces
Summary
Valuations on convex bodies can be considered as valuations on suitable function spaces. Wang and Liu [55] showed that the Fourier transform is the only valuation which is a continuous, positive GL(n) covariant and logarithmic translation covariant complex-valued valuation on integral functions. Let Lp (C, Rn ) denote the Lp -space of Lebesgue measurable complex-valued functions on Rn. for all f Lp (C,Rn ). Lp (C, S n 1) denote the Lp -space of spherical Lebesgue measurable complex-valued functions on S n 1.
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