We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.

翻译：本文探讨了将多边形划分为尽可能“圆形”的块体的问题实例，即每个块体的纵横比接近1。多边形的纵横比定义为最小外接圆直径与最大内切圆直径之比。即使对于将正多边形划分为凸块体这一特定问题（本文的研究重点），该问题也颇具深度。我们证明，等边三角形的最优（最圆形）划分具有无限多个块体，且通过特定的有限划分可以任意精度逼近下界。对于五边形及所有边数k>5的正k边形，未划分的多边形本身即为最优。正方形则呈现出一种有趣的中间情况：单块划分并非最优，但平凡下界亦不可达。通过几种较为复杂的划分方案，我们将最优纵横比的范围缩小至0.01082的纵横比间隙内。