Borell–Brascamp–Lieb Inequality
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In mathematics, the Borell–Brascamp–Lieb inequality is an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
inequality due to many different mathematicians but named after Christer Borell,
Herm Jan Brascamp Herm ( Guernésiais: , ultimately from Old Norse 'arm', due to the shape of the island, or Old French 'hermit') is one of the Channel Islands and part of the Parish of St Peter Port in the Bailiwick of Guernsey. It is located in the Eng ...
and
Elliott Lieb Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis. Lieb is ...
. The result was proved for ''p'' > 0 by Henstock and Macbeath in 1953. The case ''p'' = 0 is known as the Prékopa–Leindler inequality and was re-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell–Brascamp–Lieb inequality" is due to Cordero-Erausquin,
McCann McCann may refer to: * McCann (surname) * McCann (company), advertising agency * McCann Worldgroup, network of marketing and advertising agencies * Marist College athletic facilities ** McCann Arena McCann Arena is a 3,200-seat multi-purpose ar ...
and Schmuckenschläger, who in 2001 generalized the result to
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
s such as the
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
and
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
.


Statement of the inequality in R''n''

Let 0 < ''λ'' < 1, let −1 / ''n'' ≤ ''p'' ≤ +∞, and let ''f'', ''g'', ''h'' : R''n'' → [0, +∞) be integrable functions such that, for all ''x'' and ''y'' in R''n'', :h \left( (1 - \lambda) x + \lambda y \right) \geq M_ \left( f(x), g(y), \lambda \right), where : \begin M_ (a, b, \lambda) = \begin &\left( (1 - \lambda) a^ + \lambda b^ \right)^ \; \quad \text \quad ab\neq 0\\ &0 \quad \text \quad ab=0 \end \end and M_(a,b,\lambda) = a^b^. Then :\int_ h(x) \, \mathrm x \geq M_ \left( \int_ f(x) \, \mathrm x, \int_ g(x) \, \mathrm x, \lambda \right). (When ''p'' = −1 / ''n'', the convention is to take ''p'' / (''n'' ''p'' + 1) to be −∞; when ''p'' = +∞, it is taken to be 1 / ''n''.)


References

* * * * * {{DEFAULTSORT:Borell-Brascamp-Lieb inequality Geometric inequalities Integral geometry