# Volume

**Volume**, ** n-dimensional volume**, or

**refers generally to methods for assigning nonnegative real numbers to sets of points in**

*n*-volume*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.^{[1]} 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."