仿射集合凸集

首先,看看仿射集和凸集的定义:

  • A set C is affine if the line through any two distinct points in C lies in C. 
  • A set C is convex if the line segment between any two points in C lies in C.

倘若没有注意到红色部分,上面的定义看起来非常相似。但就是这么一丁点文字上的不同,却带来了截然不同的东西。请看下面两个命题:

  1. Any line is affine. If it passed through zero, it is a subspace, hence also a convex cone.
  2. A line segment is convex, but not affine (unless it reduces to a point).

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