# 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 $$X$$, $$Y$$

1. their length $$\le 1$$
2. 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.

Date: 2021-03-10 Wed 00:00