通过积分几何不变量识别一般多边形区域
Identification of generic polygonal domains by integral-geometric invariants
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中文总结 AI 辅助
该研究利用点间距离分布、里斯能量函数等积分几何不变量，实现一般多边形区域的唯一识别，还补充了正多边形的识别结果，扩展了经典重建定理的适用范围。
中文摘要 AI 辅助
我们研究从非局部积分几何不变量重建平面多边形区域的问题，这些不变量包括点间距离分布、里斯能量函数（Brylinski β函数），以及凸情形下的弦长分布。我们证明，一般多边形区域可由其里斯能量函数唯一确定（至多相差欧几里得等距变换），等价于由其点间距离分布确定；对于凸区域，这也可由弦长分布重建，将Waksman的经典一般重建定理扩展到凸集之外的情形。证明使用了由外单位法向量的标量积加权的边界点间距离分布，该加权分布与里斯能量函数等价。通过分析其在临界长度处二阶和三阶导数的跳跃项与爆破项，我们可恢复多边形的边长、其循环关联关系及外角。附录中，我们针对正多边形区域，在竞争区域为凸集或边数相同的情形下，得到了补充识别结果。
英文摘要
We study a reconstruction problem of planar domains from non-local integral-geometric invariants. We show that a generic polygonal domain, not necessarily convex, is uniquely determined, up to Euclidean isometry, by the interpoint distance distribution (IDD), which, for convex domains, is equivalent to the chord length distribution. Using a boundary representation of the Riesz energy function, we replace the IDD of the domain by an equivalent boundary IDD weighted by the scalar product of the outer unit normals, which we call $ν$-weighted IDD of the boundary. It enables us to reduce the problem to a one-dimensional problem. By analyzing jumps and blow-up terms of the second and third derivatives of the $ν$-weighted IDD of the boundary, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. This can be considered as extending Waksman's classical generic reconstruction theorem for convex polygons to non-convex setting, as well as extending generic polygonal reconstruction from directional covariogram data to a one-dimensional invariant in which directional information has been integrated out.
