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Added condition that $P$ is full-dimensional.
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Given aan $n$-dimensional star-convex polyhedron $P$$P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the boundary of $P$, i.e, $\partial K = \partial P$?

Given a star-convex polyhedron $P$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the boundary of $P$, i.e, $\partial K = \partial P$?

Given an $n$-dimensional star-convex polyhedron $P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the boundary of $P$, i.e, $\partial K = \partial P$?

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Regular triangulations of star-convex polyhedra with given boundary

Given a star-convex polyhedron $P$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the boundary of $P$, i.e, $\partial K = \partial P$?