I would like to find a general definition of the volume for a full dimensional polytope in $R^n$. Could anyone give me a hint please!
Thank a lot
2026-03-26 20:44:12.1774557852
a general definition of the volume of a high dimensional polytope
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The definition is the same as for any measurable subset of $\mathbb{R}^n$, it's the Lebesgue measure of polytope's interior. More elementary definition can use the multiple Riemann integral of the polytope's chracteristic function instead, that would correspond to approximating the polytope by small boxes and taking the limit. In dimensions higher than two it is not generally possible to split an arbitrary polytope into simplices (like a polygon can be split into triangles) to get a definition without limits, but it is possible for generic convex polytopes, see also here.
For ways to compute the volume see this and this related questions.