Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.

Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of vectors of E to the norm of their exterior product $\|x_1\wedge\cdots\wedge x_k\|$.

I'm interested in the measure of the set of vectors having their image by $v_k$ bounded by a constant, i.e. to the value of the integral: $$ \int_{(\mathbb{R}^n)^k} \mathbb{1}[{v_k}<B] d\mu^{\otimes k}, $$ for any $B>0$.

My try has been to write the value of $\|x_1\wedge\cdots\wedge x_k\|^2$ as a determinant and develop it by rows to find an inductive formula, the base case ($k=1$) being particularly easy, as being the volume of the Euclidean ball of radius $B$... but the computation becomes hideous.

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.

Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of vectors of E to the norm of their exterior product $\|x_1\wedge\cdots\wedge x_k\|$.

I'm interested in the measure of the set of vectors of bounded norms having their image by $v_k$ bounded by constant, i.e. to the value of the integral: $$ \int_{(\mathcal{B}_0(r))^k} \mathbb{1}[{v_k}<B] d\mu^{\otimes k}, $$ for any $r, B>0$, and where $\mathcal{B}_0(r)$ is the Euclidean ball of radius $r$ centred on $0$.

My try has been to write the value of $\|x_1\wedge\cdots\wedge x_k\|^2$ as a determinant and develop it by rows to find an inductive formula, the base case ($k=1$) being particularly easy, as being the volume of the Euclidean ball of radius $B$... but the computation becomes hideous.