Layer cake representation

In mathematics, the layer cake representation of a non-negative, real-valued measurable function f defined on n-dimensional Euclidean space Rⁿ is the formula

f(x)=\int _{0}^{{+\infty }}1_{{L(f,t)}}(x)\,{\mathrm {d}}t{\text{ for all }}x\in {\mathbb {R}}^{{n}},

where 1_E denotes the indicator function of a subset E ⊆ Rⁿ and L(f, t) denotes the super-level set

L(f,t)=\{y\in {\mathbb {R}}^{n}|f(y)\geq t\}.

The layer cake representation follows easily from observing that

1_{{L(f,t)}}(x)=1_{{[0,f(x)]}}(t)

and then using the formula

f(x)=\int _{0}^{{f(x)}}{\mathrm {d}}t.

The layer cake representation takes its name from the representation of the value f(x) as the sum of contributions from the "layers" L(f, t): "layers"/values t below f(x) contribute to the integral, while values t above f(x) do not.

References

Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.

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Layer cake representation

See also

References