AI 中文总结
本文将Federer的线性投影定理扩展到非线性问题，提出统一方法研究集合的低维非线性像与例外集，并证明了1-可求长集的锚定距离集、径向投影、圆并集的相关性质。
AI 中文摘要
欧氏空间中的投影定理建立了集合的几何结构与其低维像的大小之间的基本联系。对于ℝᵈ中的1-可求长集，Federer的经典定理表明，这类集合的1维豪斯多夫测度受有限多个线性独立投影的重数加权长度控制。我们开发了一个框架，将Federer的结果扩展到多种非线性问题中。该技术为通过低维非线性像研究集合，以及研究投影行为不佳的例外集合提供了统一方法。作为该技术的示例，我们证明：(i)每个1-可求长集都包含一个锚点，其锚定距离集具有正勒贝格测度，而不满足该性质的锚点构成的例外集包含在一个(d-2)维仿射子空间中；(ii)若1-可求长集不是本质线性的，则从至多一个有利位置出发，其平面径向投影的长度可不为正；(iii)在半径函数满足温和假设的情况下，以1-可求长集为中心的圆的并集具有正面积。
英文摘要
Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well as studying the exceptional sets which exhibit poor projective behavior. As illustrations of our technique, we show that (i) every 1-rectifiable set contains a pin whose pinned distance set has positive Lebesgue measure, and that the exceptional set of pins for which this fails is contained in a $(d-2)$-dimensional affine subspace; (ii) planar radial projections of a 1-rectifiable set can fail to have positive length from at most one vantage point unless the set is essentially linear; and finally (iii) unions of circles centered on a 1-rectifiable set have positive area under mild assumptions on the radius function.
Comments17 pages
