AI 中文总结
该研究在给定对称约束下，分类了固定原点的有限等距变换群G作用下不变的常宽度凸体中的最小面积者，明确了不同G对应的极小值类型。
AI 中文摘要
经典的Blaschke-Lebesgue定理指出，Reuleaux三角形是具有常宽度的平面凸体中面积最小的。我们在给定对称约束下研究该极值问题，具体分类了在固定原点的有限等距变换群G作用下不变的常宽度凸体中的最小面积者。对于特殊反射群D₁，极小值恰好是在给定反射下不变的Reuleaux三角形；若G包含半转R_π，则圆盘是唯一极小值；对于奇数n≥3，极小值为正Reuleaux n边形，在循环群Cₙ情形下旋转后唯一，在二面体群Dₙ情形下恰好是满足给定反射对称的那些。
英文摘要
The classical Blaschke--Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_π$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.
Comments19 pages, 2 figures
