Logvolumen

Logvolumen, or log-volume, is a term used in mathematics and related disciplines to denote the natural logarithm of the geometric volume of a region in Euclidean space. If A is a measurable subset of R^d with finite positive d-dimensional volume Vol_d(A), the log-volume is defined as log Vol_d(A). The base is usually the natural base e, though other bases may be used in different contexts. In purely mathematical treatments volumes are often treated as dimensionless numbers by choosing a standard unit of length.

Basic properties include translation invariance and scaling behavior. Since Vol_d(A + x) = Vol_d(A), the log-volume is unchanged under translations. If a linear map T is applied, Vol_d(T(A)) = |det T| Vol_d(A); hence the log-volume shifts by log |det T|. For unions, Vol_d(A ∪ B) equals Vol_d(A) + Vol_d(B) minus the overlap, so there is no simple additive rule for log-volumes in general (log Vol(A ∪ B) is not simply related to log Vol(A) and log Vol(B)).

Common examples illustrate the concept. The unit cube [0,1]^d has Vol = 1, so its log-volume is 0. The unit ball B^d has Vol = π^{d/2} / Γ(d/2 + 1), and its log-volume is ln(π^{d/2} / Γ(d/2 + 1)).

Computation in practice often relies on known exact formulas for simple shapes or numerical methods for irregular regions, such as Monte Carlo integration or voxelization. Log-volume serves as a scalar descriptor of the size of a region and is used in high-dimensional geometry and related fields to compare volumes after normalization or to study asymptotic growth.