= Solution
The <open star in a simplicial complex> of a <vertex> $v$ is the <union> of the relative interiors of the simplices containing $v$. In barycentric coordinates it is
$$
\boxed{\operatorname{st}(v)=\{p\in|K|:t_v(p)>0\}.}
$$
Let $L$ be the <subcomplex> consisting of all simplices not containing $v$. The <complement> of the star is exactly $|L|$: the positive-coordinate support of a point misses $v$ precisely when it is a <simplex> of $L$. By the allowed closed-subcomplex theorem, $|L|$ is <closed> in $|K|$. Hence the star is <open>.
This is the open-star convention required by the question. The <closed> star, which also includes the opposite faces, is a different subset and need not be <open>.
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