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Carathéodory’s theorem - mklabgr - 06-19-2026
Down in the depths ‘on’ Carathéodory’s theorem
By David Orden
Summary
The article “Down in the depths ‘on’ Carathéodory’s theorem” explains a modern extension of the classical theorem proved by Constantin Carathéodory in 1907, a cornerstone of convex geometry. The original theorem states that any point inside the convex hull of a set in (d)-dimensional space can be represented using at most (d+1) points from that set.
The article focuses on recent work by Ruy Fabila-Monroy and Clemens Huemer, who introduced a “depth” version of the theorem based on Tukey depth, a measure of how deeply a point lies inside a point set. Their result shows that the deeper a point is within the convex hull, the more ways it can be enclosed by large groups of points, providing a richer geometric understanding than the classical theorem. The work also extends related results such as Helly’s and Kirchberger’s theorems, offering a new perspective on fundamental ideas in discrete and computational geometry.
ARTICLE