凸集与严格凸集的区别解析及ℝ/ℝ²/ℝ³中严格凸集实例请求

Let’s start with a quick anchor to ground this: a convex set means any line segment between two points in the set lies entirely within the set. A strictly convex set adds a tighter rule: for any two distinct points in the set, every point on their connecting line segment (except the endpoints) sits in the interior of the set. No flat edges or straight boundary segments allowed!

1D (ℝ) #

• Convex but not strictly convex: Closed intervals like [0, 1]. Grab the endpoints 0 and 1—every point on the segment between them is on the interval’s boundary, not its interior. That’s why it misses strict convexity.
• Strictly convex sets:
  • Open intervals like (0, 1): Pick any two distinct points a, b ∈ (0,1) and a weight t ∈ (0,1)—the point ta + (1-t)b is strictly between 0 and 1, so it’s an interior point of the interval.
  • Unbounded open sets like (0, ∞): Any weighted average of two positive numbers is positive and never touches the "boundary" (since 0 isn’t in the set, and there’s no upper limit).

2D (ℝ²) #

• Convex but not strictly convex:
  • Closed squares (e.g., {(x,y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}): Pick two points on the top edge, like (0,1) and (1,1). The entire segment between them lies on the square’s boundary, not its interior.
  • Closed disks (e.g., {(x,y) | x² + y² ≤ 1}): Take diametrically opposite points like (1,0) and (-1,0)—their connecting segment is the disk’s diameter, which sits entirely on the boundary.
• Strictly convex sets:
  • Open disks (e.g., {(x,y) | x² + y² < 1}): Any two distinct points inside the disk—all interior points of their connecting segment stay strictly inside the disk (never touching the boundary).
  • Open elliptical regions (e.g., {(x,y) | x²/4 + y²/9 < 1}): The curved boundary ensures no straight segments exist on the edge, so every interior point of a segment between two set points stays inside the ellipse.

3D (ℝ³) #

• Convex but not strictly convex:
  • Closed cubes (e.g., {(x,y,z) | 0 ≤ x,y,z ≤ 1}): Any two points on a face’s edge will have their connecting segment sitting on the cube’s boundary.
  • Closed spheres (e.g., {(x,y,z) | x² + y² + z² ≤ 1}): A line segment between two points on the sphere’s surface lies entirely on that surface, not in the interior.
• Strictly convex sets:
  • Open spheres (e.g., {(x,y,z) | x² + y² + z² < 1}): Just like the 2D open disk, all interior points of segments between distinct set points are strictly inside the sphere.
  • Open ellipsoids (e.g., {(x,y,z) | x² + y²/4 + z²/9 < 1}): The curved, non-planar boundary guarantees no straight boundary segments, checking all the boxes for strict convexity.

The big visual takeaway: strictly convex sets have smooth, rounded boundaries with no flat edges, while convex-but-not-strictly-convex sets have at least one straight segment on their edge.

内容的提问来源于stack exchange，提问作者JustANoob