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.

See also

• Symmetric decreasing rearrangement

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.