The most inaccessible point of a convex domain

TL;DR

This paper investigates the concept of inaccessibility points within convex domains, characterizing their structure and properties, especially in strictly convex shapes and polygons like triangles, revealing new geometric insights.

Contribution

It provides a detailed analysis of the maximum inaccessibility points in convex domains, showing their possible configurations and distinguishing features in polygons and triangles.

Findings

In strictly convex domains, I_D is either a point or a segment.

In planar polygons, I_D is generally a point.

The inaccessibility point in a triangle is not a classical notable point.

Abstract

The inaccessibility of a point p in a bounded domain D \subset R^n is the minimum of the lengths of segments through p with boundary at \bd D. The points of maximum inaccessibility I_D are those where the inaccessibility achieves its maximum. We prove that for strictly convex domains, I_D is either a point or a segment, and that for a planar polygon I_D is in general a point. We study the case of a triangle, showing that this point is not any of the classical notable points.