Minimum-Weight Steiner Triangulation of Convex Polygons Requires Interior Steiner Points

2026-06-23

AI Summary

This study addresses a long-standing conjecture by Eppstein (1994), which posited that the minimum-weight Steiner triangulation of any convex polygon can be achieved using only boundary Steiner points. The authors construct the first counterexample demonstrating that, for certain convex polygons, the optimal triangulation necessarily requires interior Steiner points. By leveraging tools from computational geometry and Steiner triangulation theory, they design a specific convex polygon instance and rigorously prove that its minimum-weight triangulation depends on the inclusion of internal Steiner points. This result not only refutes the sufficiency of boundary-only Steiner points but also expands the theoretical understanding of minimum-weight triangulation problems, offering new insights into their algorithmic complexity and structural properties.

Abstract

We construct a convex polygon for which the minimum-weight Steiner triangulation requires an interior Steiner point. This provides a counterexample to a 1994 conjecture of Eppstein that minimum-weight Steiner triangulation of convex polygons needs only Steiner points on the boundary of the polygon.

Problem

• Steiner triangulation
• convex polygon
• minimum-weight
• interior Steiner point
• computational geometry

Innovation

• Steiner triangulation
• minimum-weight triangulation
• convex polygon
• interior Steiner point
• computational geometry