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Polyhedral Horofunction Compactification as a Polyhedral Ball

Lizhen Ji, Anna-Sofie Schilling

Abstract

In this paper we answer positively a question raised by Kapovich and Leeb in a paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, we show that for a finite-dimensional vector space with a polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this, we establish a criterion for converging sequences in the horofunction compactification and generalize the basic notion of the moment map in the theory of toric varieties.

Polyhedral Horofunction Compactification as a Polyhedral Ball

Abstract

In this paper we answer positively a question raised by Kapovich and Leeb in a paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, we show that for a finite-dimensional vector space with a polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this, we establish a criterion for converging sequences in the horofunction compactification and generalize the basic notion of the moment map in the theory of toric varieties.

Paper Structure

This paper contains 14 sections, 20 theorems, 70 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Cones, subspaces and convex hulls
  5. Some convex analysis
  6. The "pseudo-norm" $\lvert \cdot \rvert_R$
  7. The maps $h_{E,p}$
  8. The horofunction compactification
  9. Introduction to horofunctions
  10. The characterization theorem
  11. Examples
  12. Proof of Theorem \ref{['thm:homeo']}
  13. The map $m^C$
  14. The actual proof of Theorem \ref{['thm:homeo']}

Key Result

Lemma 2.7

Let $E \subset B^{\circ} \subset X^*$ be a face and $F \subset B \subset X$ its dual. Then there is a point $t \in V(F)^* \subseteq X^*$ such that for all $e \in E$ and $q \in V(F)$.

Figures (6)

  • Figure 1: Schematic picture to illustrate the notations
  • Figure 2: The unit ball $B$ and its dual $B^{\circ}$ as in Example \ref{['ex:dualL1']}
  • Figure 3: Sketch of the idea of constructing new unit balls.
  • Figure 4: left: A schematic picture of the cone $K_F$ with $K_{F_1}$ in its boundary. right: The projections of points with $y^{F_1}$ bounded remain within bounded distance to $\partial_{\text{rel}} K_{F_1}$ (view from the origin into $K_F$).
  • Figure 5: The unit ball $B$ (left) and its dual $B^{\circ}$ (right) with some faces. The face $F_1$ corresponds to the point $E_1$ while $F_2$ is dual to the facet $E_2$.
  • ...and 1 more figures

Theorems & Definitions (57)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 2.6: hsw, Lemma 3.8
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.9
  • ...and 47 more