教育与艺术发展

从星形线切线性质到挂谷猜想：定长线段旋转的面积压缩之旅

吕 川

中国石油大学 ( 华东 ) 理学院

吴 瑞华

中国石油大学 ( 华东 ) 理学院

DOI: https://doi.org/10.59429/jyys.v3i4.13964

Keywords: 星形线；切线；挂谷集

---

Abstract

本文从本科微积分中经典的星形线切线性质出发，首先证明其第一象限内任意切线被坐标轴截得线段为定值的核心性质，揭示其作为定长线段 360°旋转包络线的特殊情况；通过与三尖瓣线、佩龙树等构造的面积对比，展现分形方法对旋转区域的压缩潜力；最终自然引出贝西科维奇集与挂谷猜想，介绍其测度性质、与调和分析的联系以及 2025 年三维挂谷猜想的重大突破，为微积分教学提供一个连接经典知识与前沿研究的典型案例。

---

References

[1] Yates, R. C. (1974). Curves and Their Properties [M]. Reston, VA: National Council of Teachers of Mathematics.

[2] Perron, O. (1928). Über einen Satz fon Besicovitsch [J]. Mathematische Zeitschrift, 28(1), 383-386.

[3] Falconer, K. J. (1986). The Geometry of Fractal Sets [M]. Cambridge University Press.

[4] Besicobitch, A. S. (1928). On the fundamental geometrical properties of linearly measurable plane sets of points [J]. Mathematische Annalen,98:422-464. https://doi.org/10.1007/BF01451603.

[5] Wolff, T. (1999). Recent work connected with the Kakeya problem [C]. In Prospects in mathematics (pp. 129-162). Providence, RI: American Mathematical Society.

[6] Stein, E. M. (1993). Harmonic Analysis: Real-VariableMethods, Orthogonality, and Oscillatory Integrals [M]. Princeton: Princeton University Press.

[7] Fefferman, C. (1971). The multiplier problem for the ball [J]. Annals of Mathematics, 94(2), 330-336. https://doi.org/10.2307/1970864.

[8] Wang, H., Zahl, J. (2025). Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions [EB/OL]. arXiv preprint arXiv:2502.17655.

[9] Tao, T. (2025, February 25). The three-dimensional Kakeya conjecture, after Wang and Zahl[EB/OL].https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/.