This paper aims to characterize the optimal frame for phase retrieval, defined as the frame whose condition number for phase retrieval attains its minimal value. In the context of the two-dimensional real case, we reveal the connection between optimal frames for phase retrieval and the perimeter-maximizing isodiametric problem, originally proposed by Reinhardt in 1922. Our work establishes that every optimal solution to the perimeter-maximizing isodiametric problem inherently leads to an optimal frame in ${\mathbb R}^2$. By recasting the optimal polygons problem as one concerning the discrepancy of roots of unity, we characterize all optimal polygons. Building upon this connection, we then characterize all optimal frames with $m$ vectors in ${\mathbb R}^2$ for phase retrieval when $m \geq 3$ has an odd factor. As a key corollary, we show that the harmonic frame $E_m \subset {\mathbb R}^2$ is {\em not} optimal for any even integer $m \geq 4$. This finding disproves a conjecture proposed by Xia, Xu, and Xu [{\em Math. Comp.}, 94 (2025), pp.~2931--2960]. Previous work has established that $E_m$ is indeed optimal when $m$ is an odd integer.

翻译：本文旨在刻画相位恢复的最优框架，即其相位恢复条件数达到最小值的框架。在二维实情形下，我们揭示了相位恢复最优框架与周长最大化等径问题之间的关联，后者最初由Reinhardt于1922年提出。我们的研究证明：周长最大化等径问题的每个最优解必然导出${\mathbb R}^2$中的一个最优框架。通过将最优多边形问题重构为单位根偏差问题，我们完整刻画了所有最优多边形。基于这一关联性，我们进一步刻画了当$m \geq 3$含有奇因子时，${\mathbb R}^2$中具有$m$个向量的相位恢复最优框架。作为一个关键推论，我们证明对于任意偶数$m \geq 4$，调和框架$E_m \subset {\mathbb R}^2$均非最优。这一发现否定了Xia、Xu和Xu在[{\em Math. Comp.}, 94 (2025), pp.~2931--2960]中提出的猜想。先前研究已证实当$m$为奇数时，$E_m$确实是最优框架。