# Characterisation of a superset of the simplex

Does there exist a nice description of the following set: $$$$A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace,$$$$ where $$\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 $$j \ne i$$.