Does there exist a nice description of the following set: \begin{equation} A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace, \end{equation} where $\bar{x}$ is the arithmetic mean $\bar{x}=\frac{1}{n}x\cdot\mathbf{1}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$ ? Here by 'nice' I mean anything which provides a better understanding of the set, for instance in terms of statistics and/or geometry. Of course, if we denote by $\tilde{\Delta}:=\lbrace x\in\mathbb{R}^{n}\ \colon\ x\cdot\mathbf{1}=1\rbrace$, we have $A=\operatorname{pr}^{-1}_{\tilde{\Delta}}(( 0,1)^{n})$, but that's not helping for intuition. Clearly $A$ is a superset for the unit-simplex $\Delta=\lbrace x\in\mathbb{R}_{+}^{d}\ \colon x\cdot\mathbf{1} = 1\rbrace$. However, it may also contain negative numbers to some extend, as one can easly check for $n=2$, by considering e.g. $(-\frac{1}{2},-\frac{1}{3})$. I'm thankful for any suggestions.

The boundary of your region lies in the hyperplanes $$ x_i = \overline{x} - \frac{1}{n}$$ and $$ x_i = \overline{x} + \frac{n-1}{n}$$

Note that the region is invariant under translation by $(1,\ldots, 1)$. Its cross-sections by the hyperplanes $\overline{x} = c$ are simplices, whose extreme points are of the form $x_i = c+(n-1)/n$ for one $i$, $x_j = c-1/n$ for $j \ne i$.