Erdős–Nagy theorem

Every simple polygon can be transformed into a convex polygon by finitely many reflections of its pockets across their diagonals, without changing edge lengths.

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Polygon pocket: A maximal region of a simple polygon lying outside its convex hull, bounded by a polygonal chain and a hull edge. Each step reflects one such inward indentation across its diagonal.
Paul Erdős: A Hungarian mathematician known for prolific collaborations and contributions across discrete mathematics and number theory. He posed the polygon-convexification problem that led to the theorem.
Polygon: A plane figure bounded by a finite closed chain of straight line segments. The theorem concerns a particular transformation of polygons.
Polygonal linkage: A framework of rigid bars joined at flexible hinges, often represented by a polygon’s edges. The theorem shows that a polygonal chain can change shape while retaining all side lengths.
Diagonal of a polygon: A line segment joining two nonconsecutive vertices of a polygon. The pocket’s diagonal is the axis across which the pocket is reflected.
Béla de Szőkefalvi-Nagy: A Hungarian mathematician whose work included geometry, functional analysis, and operator theory. He proved that repeated pocket reflections always reach a convex polygon.
Convex set: A set in which the line segment between any two points of the set lies entirely within it. Convexity is the defining property of the theorem’s final polygon.
Carpenter's rule problem: The problem of continuously straightening a simple polygonal chain without crossings. It extends the theme of changing polygonal shape while preserving edge lengths.
Convex hull: The smallest convex set containing a given set of points. Pockets are defined relative to the polygon’s convex hull.
Polygon convexification problem: The problem of transforming a nonconvex polygon into a convex one under specified geometric constraints. The theorem gives a finite solution using reflections that preserve side lengths.