I never worked with $n$ dimensional geometry or the measure theory, but now I need a solution to the following problem. I have $m$ vectors from $R^n$ and need to compute measure of the figure that they form.
Concretely, if $n=2$, I would be computing the area of the corresponding polygon, which is something I know how to do. If $n=3$, I would be looking for the volume. Also, if $n=3$ and the points of the given vectors are in the same plane, I'd be computing the corresponding area. However, if $n > 3$, I really don't know what to do and how is the measure I'm looking for called. I guess that it could have something to do with integrals, but I'm not sure. For the record, I never did integrals for functions of more than one argument.
Here is the more formal problem statement (mathematically correct wording is probably different, but as I stated above, I'm completely unfamiliar with this material):
Let $X^{(i)} = [x_1^{(i)}, x_2^{(i)}, \ldots, x_n^{(i)}]$, $i=1,2,\ldots,m$. Compute measure of the figure whose vertices are the points determined by the vectors $X^{(1)}, X^{(2)}, \ldots, X^{(m)}$.
Thanks in advance. :)
What about splitting the figure into non-overlapping $n$-simplices (a higher dimensional version of triangles) with vertices among the vertices of the given figure, computing the $n$-volume of each of them and then summing it?
Assume your figure is an $n$-dimesnional polytope. Separate it into $n$-simplices so that the vertices of each simplex are also vertices of the whole figure. The $n$-volume of a simplex with vertices (there are always $n+1$ vertices of an $n$-simplex) $X^{(j_0)}, \, X^{(j_1)},..., \, X^{(j_{n})}$ is computed as the following determinant $$V_{n} = \left| \frac{1}{n!} \det\Big(X^{(j_1)} - X^{(j_0)}, \, \,\, X^{(j_2)} - X^{(j_0)}, \,\,\, X^{(j_3)} - X^{(j_0)}, \,\, \, ..., \,\,\,\ X^{(j_n)} - X^{(j_0)} \Big) \right|.$$
This is the same as having a complicated polygon and then splitting it into triangles with vertices belonging to the set of vertices of the polygon (basically drawing diagonals with disjoint interors until your polygon is fully triangulated).
Are your figures convex or more general? Are they allowed to self intersect or not?