In
mathematics, a
Borel measure ''μ'' on ''n''-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
is called logarithmically concave (or log-concave for short) if, for any
compact subsets ''A'' and ''B'' of
and 0 < ''λ'' < 1, one has
:
where ''λ'' ''A'' + (1 − ''λ'') ''B'' denotes the
Minkowski sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set
: A + B = \.
Analogously, the Minkowski ...
of ''λ'' ''A'' and (1 − ''λ'') ''B''.
Examples
The
Brunn–Minkowski inequality asserts that the
Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
is also log-concave.
By a theorem of Borell,
a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a
logarithmically concave function In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality
:
f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^
for all and . If is strict ...
. Thus, any
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ...
is log-concave.
The
Prékopa–Leindler inequality In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is ...
shows that a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of log-concave measures is log-concave.
See also
*
Convex measure, a generalisation of this concept
*
Logarithmically concave function In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality
:
f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^
for all and . If is strict ...
References
{{Measure theory
Measures (measure theory)