# Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the boundaries of a Voronoi tessellation of $\mathbb{R}^3$ with respect to the points $x_i$, using the shortest-path metric induced by the obstacles? By comparison, in $\mathbb{R}^2$, it is easy to show that the boundaries of a Voronoi tesselation are hyperbolic arcs.

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