I am trying to compute the volume of the following space to get an estimate on some combinatorial object. I have a regular $n$-simplex embedded in $\mathbb{R}^{n+1}$ by $$ \Delta_n := \left\{ (x_0,\dots,x_n)\in\mathbb{R}^{n+1} :~ x_i\ge 0,~ \sum_{i=0}^n x_i = 1\right\} $$ I would like to compute the volume of a ball of radius $1$ in $\Delta_n$ centered at $\left(\frac{1}{n+1},\dots,\frac{1}{n+1}\right)$ in the $\ell_1$-norm. i.e. the volume of
$$\left\{ (x_0,\dots,x_n)\in\mathbb{R}^{n+1}:~ x_i\ge 0, ~\sum_{i=0}^n x_i=1, ~\sum_{i=0}^n \left|x_i - \frac{1}{n+1} \right| \le 1\right\} $$
I mainly care about the ratio of this volume to the volume of $\Delta_n$ so any estimate in that direction would be very appreciated. I've been embarrassingly banging my head against the wall on this one for 2 weeks.