In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
''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 coo ...
al
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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 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'':
A + B = \
The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...
of ''λ'' ''A'' and (1 − ''λ'') ''B''.
Examples
The
Brunn–Minkowski inequality asserts that the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
is log-concave. The restriction of the Lebesgue measure to any
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
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 stri ...
. Thus, any
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
is log-concave.
The
Prékopa–Leindler inequality shows that a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
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 stri ...
References
{{Measure theory
Measures (measure theory)