几何分析讨论班

报告题目：k-Medial Axes, Dimension, and the Erdős Conjecture

报 告 人：梁湘玉 教授（北京航空航天大学）

时  间：2026年7月2日（星期四），上午10:00-11:00

地  点：海纳苑2幢204

摘  要：The medial axis of a planar region refers to the set of all points within the region whose distance to the boundary is realized by two distinct boundary points simultaneously. When the boundary of the region is piecewise analytic, the medial axis retains many topological and geometric properties of the region while having a lower dimension, making it suitable for storage and widely studied in the field of computer imaging.

More generally, in higher-dimensional spaces, the set of points whose distance to the boundary is realized by k linearly independent boundary points is called the k-medial axis of the region. In 1945, P.Erdős discussed the dimension of k-medial axes. He proved that for any region in n-dimensional Euclidean space, the medial axis is a set of measure zero, and the (n+1)-medial axis is a countable set. He further conjectured that for any region in n-dimensional Euclidean space, the Hausdorff dimension of the k-medial axis is at most n - k + 1.

We will introduce the fundamental properties of k-medial axes and prove the aforementioned Erdős conjecture.