Compute the volume of $$ S_n=\{(x_1,x_2,...,x_n)\in\mathbb{R^n},x_i\geq 0,\displaystyle\sum_{k=0}^{n} x_i<1\} $$
I don't really have an idea how to solve it.
My 'work': Perhaps I could use $$ ||x||_1=|x_1|+...+|x_n| $$ So the set is $$ S_n=\{(x_1,x_2,...,x_n)\in\mathbb{R^n},x_i\geq 0,B(0,1)\} $$ Where $$ B(0,1)=\left\{M\in\mathbb{R^n} \,\mid\,d(M,0)<1\right\} $$ But I don't see where it leads me...
Thank you in advance
Let $V_n$ denote the (as-yet unknown) volume of $S_n$. If $S_n(t)$ denotes the simplex $$ S_n(t) = \{(x_1, \dots, x_n) : x_i \geq 0, \sum_{i=1}^n x_i \leq t\}, $$ so that $S_n(1)$ is your $S_n$, then the volume of $S_n(t)$ is $V_n t^n$. (Proof: Scale $S_n$ by a factor of $t$.)
Slicing $S_n$ by the hyperplane $x_n = t$ with $0 \leq t \leq 1$ gives (a translated copy of) $S_{n-1}(1 - t)$ as cross section. Cavalieri's theorem allows you to express $V_n$ recursively in terms of $V_{n-1}$, and this recursion is almost trivial to solve in closed form.
