Unit sphere is convex
When we say "A is convex", means for any two points in \(A\), the line in \(A\)(each point on the line in \(A\)).
To prove the title "Is the unit sphere convex?", we need to prove for any two vectors \(X\), \(Y\)
- their length \(\le 1\)
- their linear combination \(\le 1\)
First, their linear combination belongs to unit sphere \(S\), when \(t \in [0, 1]\) and such linear combination is \((1 - t)X + tY\).
By triangle inequality principle, we know
\[
| (1 - t)X + tY | ≤ | 1-t | X | + | t | Y |
\]
By \(||X||, ||Y|| \le 1\), we know
\[
| 1-t | X | + | t | Y | ≤ | 1-t | + | t | = 1 |
\]
Thus,
\[
| (1 - t)X + tY | ≤ 1 |
\]
Proved.