Question: A subset S C R is called convex if, for any x, y S, the line segment joining x to y is also in

A subset S C R" is called convex if, for any x, y ∈ S, the line segment joining x to y is also in S, i.e., t x + (1 - t) y ∈ S for all 0 ≤ t ≤ 1. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?

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