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Unit sphere is convex
source link: https://dannypsnl.github.io/blog/2021/03/10/math/sphere-is-convex/
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Unit sphere is convex
When we say "A is convex", means for any two points in AAA, the line in AAA(each point on the line in AAA).
To prove the title "Is the unit sphere convex?", we need to prove for any two vectors XXX, YYY
- their length ≤1\le 1≤1
- their linear combination ≤1\le 1≤1
First, their linear combination belongs to unit sphere SSS, when t∈[0,1]t \in [0, 1]t∈[0,1] and such linear combination is (1−t)X+tY(1 - t)X + tY(1−t)X+tY.
By triangle inequality principle, we know
∣∣(1−t)X+tY∣∣≤∣1−t∣∣∣X∣∣+∣t∣∣∣Y∣∣|| (1 - t)X + tY || \le |1-t| ||X|| + |t| ||Y||∣∣(1−t)X+tY∣∣≤∣1−t∣∣∣X∣∣+∣t∣∣∣Y∣∣
By ∣∣X∣∣,∣∣Y∣∣≤1||X||, ||Y|| \le 1∣∣X∣∣,∣∣Y∣∣≤1, we know
∣1−t∣∣∣X∣∣+∣t∣∣∣Y∣∣≤∣1−t∣+∣t∣=1|1-t| ||X|| + |t| ||Y|| \le |1-t| + |t| = 1∣1−t∣∣∣X∣∣+∣t∣∣∣Y∣∣≤∣1−t∣+∣t∣=1
Thus,
∣∣(1−t)X+tY∣∣≤1|| (1 - t)X + tY || \le 1∣∣(1−t)X+tY∣∣≤1
Proved.
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