Uniqueness of Maximal Inscribed Parabolas and Minimal Circumscribing Horocycles

2025/09/22 by Lukarevski, Martin, Schröcker, Hans-Peter · 1 citation
#51M16 52A40 51M04 51M09 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2509.17415

Abstract

We prove existence of three unique ``exparabolas'' to a triangle. Each of these parabolas is internally tangent to one edge and the two other sides. Among all like parabolas, it is characterized by having maximal parameter. We use this result to prove a more general uniqueness statement on maximal parabolas into a convex point set. In similar spirit, we demonstrate uniqueness of minimal enclosing horocycles in hyperbolic geometry, provided the enclosed set is sufficiently small.

Citations

• On the Löwner-John Ellipsoids of the Metric Polytope
• Simple Approximations of Semialgebraic Sets and their Applications to Control
• Minimal area conics in the elliptic plane

Cited by

• Exparabolas of a Triangle

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