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Power diagrams and Morse Theory

Martijn van Manen·Dirk Siersma
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Abstract

We study Morse theory of the (power) distance function to a set of points in $\mathbb{R}^n$. We describe the topology of the union of the corresponding set of growing balls by a Morse poset. The Morse poset is related to the power tesselation of $\mathbb{R}^n$. We remark that the power diagrams from computer science are the spines of amoebas in algebraic geometry, or the hypersurfaces in tropical geometry. We show that there exists a discrete Morse function on the coherent triangulation, dual to the power diagram, such that its critical set equals the Morse poset of the power diagram.

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Martijn van Manen, Dirk Siersma. 2005-08-01. Power diagrams and Morse Theory. https://arxiv.org/abs/math/0508037

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