Polyhedral combinatorics of bisectors

TL;DR

This work develops a polyhedral framework for bisectors under polyhedral norms by introducing bisection cones and bisection fans, revealing how the presence of maximal cells in a bisector depends on the relative site direction. It proves the existence of bisection fans in two dimensions and provides explicit constructions for the $\ell_{1}$- and $\ell_{\infty}$-norms as well as the discrete Wasserstein distance, with intricate structures such as the resonance arrangement appearing in the Wasserstein case. The authors show that bisection fans refine the common refinements of the face fan and a facet-hyperplane arrangement, and they derive sharp complexity bounds for the number of maximal cells in bisectors across several norm families. The results illuminate fundamental combinatorial patterns in polyhedral Voronoi-type diagrams and open questions about higher-dimensional existence and structural criteria, with implications for geometric combinatorics and metric geometry.

Abstract

For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the $\ell_{1}$-norm and the $\ell_{\infty}$-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.

Figures (11)

• Figure 1: The unit ball $P$ of the $\ell_{1}$-norm with facets labelled $1,2,3,4$ with thickened edges. The bisector $\mathop{\mathrm{bis}}\limits^{P}(0, a)$ is marked in blue. The point $x$ lies in $\mathop{\mathrm{bis}}\limits^{P}(0, a)$ since the red, scaled unit ball centered at $x$ contains $0$ and $a$ simultaneously in the boundary. The shaded region is $C_{1} \cap (C_{4}+a)$, the intersection of the cones over the facet $1$ and $4$ with apices $0$ and $a$, respectively. The maximal cell $\mathop{\mathrm{bis}}\limits_{1, 4}(0, a)$ is contained in this region.
• Figure 2: A $2$-dimensional bisector of two points under the $\ell_{1}$-norm in dimension $2$. Since $a-b$ is parallel to a facet of the unit ball, $\mathop{\mathrm{bis}}\limits(a, b)$ contains full-dimensional cells.
• Figure 3: A subdivision of $H^{\geq}$ into the cones $y+K_{-G}$. Translating $y$ to the origin, this subdivision is the decomposition in Equation \ref{['eq:bis_cone_subdiv']}.
• Figure 4: The generators of $\mathcal{B}_{10, 4}$ arise from the diagonals of the quadrilateral with vertices $v_{10},v_1,v_4$ and $v_5$.
• Figure 5: The half-plane $H^{\geq}_{1,7}$ is subdivided into $5$ bisection cones $\mathcal{B}_{1, 7}$, $\mathcal{B}_{2, 8}$, $\mathcal{B}_{3, 9}$, $\mathcal{B}_{4, 10}$ and $\mathcal{B}_{5, 1}.$

Theorems & Definitions (46)

• Proposition 2.1: tropvoronoi
• Definition 2.2
• Proposition 2.3: tropvoronoi
• Theorem 2.4: tropvoronoi
• Corollary 2.5: tropvoronoi
• Definition 2.6
• Proposition 3.1
• proof
• Proposition 3.2
• proof