论文标题
定量分化和内侧轴
Quantitative differentiation and the medial axis
论文作者
论文摘要
我们从“粗糙”和“定量”的角度研究了欧几里得空间中$ k $的内侧轴（$ k $中最接近点的空间点）。我们表明，在$ k $的补充中，“大多数”球$ b（x，r）$，从$ x $中可以看出，几乎closest点的$ x $ to $ x $ in $ k $ in $ k $占据了一个小角度。换句话说，如果只有一定的有限分辨率看起来，则大多数位置和尺度在$ k $“看起来”的补充中落在内侧轴之外。 “大多数”一词涉及Carleson包装条件，我们的边界与集合$ k $无关。
We study the medial axis of a set $K$ in Euclidean space (the set of points in space with more than one closest point in $K$) from a "coarse" and "quantitative" perspective. We show that on "most" balls $B(x,r)$ in the complement of $K$, the set of almost-closest points to $x$ in $K$ takes up a small angle as seen from $x$. In other words, most locations and scales in the complement of $K$ "appear" to fall outside the medial axis if one looks with only a certain finite resolution. The word "most" involves a Carleson packing condition, and our bounds are independent of the set $K$.
