Visibility-preserving convexifications using single-vertex moves

TL;DR

This paper proves that any polygon can be convexified without losing internal visibility, and that convexification can be achieved through single-vertex moves, confirming a longstanding question in computational geometry.

Contribution

It demonstrates that the condition of using only single-vertex moves is redundant, as any visibility-preserving convexification can be done with such moves.

Findings

Any polygon can be convexified without losing internal visibility.

Single-vertex moves are sufficient for visibility-preserving convexification.

The redundancy of the single-vertex moves condition is established.

Abstract

Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using {\em single-vertex moves})? We prove the redundancy of the "single-vertex moves" condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2).