# How to compute the volume of a region transformed by a matrix?

This is a rewrite of the OP's question to emphasize what I think are the research level issues here.

Let $$\mathscr{R}$$ be a bounded convex body in $$\mathbb{R}^n$$ and let $$H : \mathbb{R}^n \to \mathbb{R}^r$$ be a surjective linear map for $$r. How can we compute the volume of $$H(\mathscr{R})$$? Of course, the answer to this question will depend on how $$\mathscr{R}$$ is given. I don't know what the OP intended, but here are some options I can see:

• $$\mathscr{R}$$ is a convex polytope, given as a list of vertices.

• $$\mathscr{R}$$ is a convex polytope, given as a list of facet inequalities

• $$\mathscr{R}$$ is a $$\{ f(x_1, \ldots, x_n) \leq c \}$$, for $$f$$ some convex polynomial. We could generalize this to $$\{ f_1 \leq c_1,\ f_2 \leq c_2,\ \cdots,\ f_N \leq c_N \}$$ for some list of convex polynomials $$f_j$$.

• There is some polynomial function $$\phi$$ sending $$\mathbb{R}^n$$ to symmetric $$k \times k$$ matrices, and $$\mathcal{R}$$ is the set of $$\vec{x}$$ so that $$\phi(\vec{x})$$ has at least $$\ell$$ nonnegative eigenvalues. (This sort of formulation is very common in semidefinite programming.)

There will probably also be different answers depending on whether we are considering $$r$$ and $$n$$ bounded, $$r$$ bounded with $$n \to \infty$$, or both $$r$$ and $$n$$ going to $$\infty$$.

The original question is below.

Consider a convex body $$\mathscr{R}\subset \mathbb{R}^n$$ and a rank-$$r$$ matrix $$\mathbf{H}=[\mathbf{h}_1,\cdots,\mathbf{h}_n]\in \mathbb{R}^{r\times n}$$. Assume that the $$r$$-dimensional volume of $$\mathbf{H}\mathscr{R}=\{\mathbf{Hr}:\mathbf{r}\in\mathscr{R}\}$$ is finite and nonzero.

How to compute it?

This problem is extended by the previous one (The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets).

• this looks like a rather standard multivariate calculus question. – Dima Pasechnik Jan 3 at 3:46
• What is "the previous one"? (Oh, I guess mathoverflow.net/questions/349554/… .) – LSpice Jan 3 at 5:11
• If $\mathcal{R}$ is a polytope given by facet inequalities, POLYMAKE polymake.org/doku.php can compute its vertices but possibly at the cost of an exponential explosion. I suspect there are smarter things to do in this case. – David E Speyer Jan 3 at 14:40
• I've taken the liberty of rewriting the question; see meta.mathoverflow.net/questions/4417 . – David E Speyer Jan 3 at 15:19
• Thanks for DES-SupportsMonicaAndTransfolk's help. The edited question is more precise and tractable. Actually, this is a problem I met in the communication theory. I am not sure the region is indeed a convex polytope. Another interesting question is how to decompose the region into a disjoint union of subregions. In the original problem, I mentioned the rank of the matrix $\mathbf{H}$, which implies that $r\le n$. It is meaningful to emphasize $n>r$. – Ryan Chen Jan 4 at 0:29