Euler measure
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In measure theory, the Euler measure of a polyhedral set equals the
Euler integral In mathematics, there are two types of Euler integral: # The ''Euler integral of the first kind'' is the beta function \mathrm(z_1,z_2) = \int_0^1t^(1-t)^\,dt = \frac # The ''Euler integral of the second kind'' is the gamma function \Gamma(z) = \ ...
of its indicator function.


The magnitude of an Euler measure

By induction, it is easy to show that independent of
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 ...
, the Euler measure of a closed bounded
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
always equals 1, while the Euler measure of a ''d''-D relative-
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
bounded convex polyhedron is (-1)^d.


See also

* Measure theory


Notes


External links


''Exponentiation and Euler measure''
{{analysis-stub Measures (measure theory) Measure theory