Volume, n-dimensional volume, or n-volume refers generally to methods for assigning nonnegative real numbers to sets of points in n-dimensional Euclidean space. It generalizes the concept of length in 1 dimension, area in 2 dimensions, and volume in three dimensions. There are no standard names for four dimensions and above, but the term hypervolume is occasionally used.
Both in general and in the case of polytopes, there are multiple definitions of n-volume designed to handle different cases. If a polytope's interior is defined and bounded, the Lebesgue measure of the interior is one way to define its n-volume.
n-volume is for most practical purposes unambiguous for finite convex polytopes. Even in this "simple" case, computing the n-volume of a convex polytope given as an intersection of half-spaces is a computationally difficult problem, barring approximations or stochastic methods. Software such as Qhull has optimized algorithms for computing the n-volume of the convex hull of a point set if n is small and the point set is not too large.
- ↑ Bárány and Füredi. "Computing the volume is difficult."