Most Circular Partition of a Square

Definitions and mathematical statement

For a compact convex plane region $K$ with nonempty interior, let $r(K)$ be the radius of its largest contained disk and $R(K)$ the radius of its smallest enclosing disk. Their centers may differ. Define its circularity ratio by $c(K)=R(K)/r(K)$.

Let $Q=[0,1]^2$. A finite convex partition is a collection $\mathcal P=\{K_1,\ldots,K_m\}$ of convex regions with nonempty interiors, pairwise disjoint interiors, and union $Q$. Its quality is the largest ratio among its pieces. Determine \[ c_* = \inf_{\mathcal P}\ \max_{1\le i\le m}c(K_i), \] where the number $m$ of pieces is not fixed. Also determine whether any finite partition attains $c_*$, and describe optimal partitions or sequences approaching the infimum.

The circles in the definition are Euclidean, and both radii are optimized separately for each piece. The objective minimizes the worst piece, not the average ratio or a ratio formed from total areas. The distinction between a minimum and an infimum matters: an improving sequence with increasing numbers of pieces need not have a finite optimal member. Rescaling the original square does not affect any circularity ratio.

Short English statement

How nearly circular can every piece of a convex partition of a square be? Find the best worst-piece ratio and decide whether finitely many pieces can achieve it.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500059 (AMR-054-0059), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 59. Original source indexed by AMR-054-0059. https://topp.openproblem.net/p59.