Least Capacity Point of Triangles

TL;DR

This paper calculates the point within an isosceles right triangle that is optimally insulated from its boundary, revisiting classical conformal mapping results to do so.

Contribution

It provides an explicit computation of the least capacity point for an isosceles right triangle, extending classical conformal mapping techniques.

Findings

Explicit formula for the least capacity point in an isosceles right triangle

Revisits and applies classical conformal mapping methods

Enhances understanding of boundary insulation in convex domains

Abstract

Let D be a compact convex domain in the plane. P\'olya & Szeg\"o and, independently, Levi & Pan defined the point p in D that is "best insulated from the boundary C of D". We compute p in the case when C is an isosceles right triangle, revisiting exact results from the study of complex conformal mappings.