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Let $k, n$ and $r$ be positive integers with $k < n$ and$r\leq\lfloor\frac{n}{k}\rfloor$. We determine the facets of the $r$-stable$n,k$-hypersimplex. As a result, it turns out that the $r$-stable$n,k$-hypersimplex has exactly $2n$ facets for every$r<\lfloor\frac{n}{k}\rfloor$. We then utilize the equations of the facets tostudy when the $r$-stable hypersimplex is Gorenstein. For every $k>0$ weidentify an infinite collection of Gorenstein $r$-stable hypersimplices,consequently expanding the collection of $r$-stable hypersimplices known tohave unimodal Ehrhart $\delta$-vectors.

Facets of the r-stable n,k-hypersimplex