A Jordan–Brouwer Separation Theorem for Polyhedral Pseudomanifolds

The Jordan Curve Theorem referring to a simple closed curve in the plane has a particularly simple proof in the case that the curve is polygonal, called the “raindrop proof”. We generalize the notion of a simple closed polygon to that of a polyhedral ( d −1)-pseudomanifold ( d ≥2) and prove a Jordan–Brouwer Separation Theorem for such a manifold embedded in ℝ d . As a by-product, we get bounds on the polygonal diameter of the interior and exterior of such a manifold which are almost tight. This puts the result within the frame of computational geometry.