Question

Show that a set is convex if and only if its intersection with any line is convex.

Answer 1

hi mate.....

So your proof is correct, but a little confusing -- especially the second part. It's easier to do directly instead of via contradiction. A little simplification:

⇐⇐: Let x,y∈Cx,y∈C and let λ∈[0,1]λ∈[0,1]. We wish to prove λx+(1−λ)y∈Cλx+(1−λ)y∈C.

Let LL be the line through xx and yy. xx and yy are in C∩LC∩L. By convexity of C∩LC∩L, we have that x+(1−λ)y∈C∩Lx+(1−λ)y∈C∩L and hence clearly x+(1−λ)y∈Cx+(1−λ)y∈C.

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