 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 isLet $L$ 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.