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June 28th, 2016, 10:39 PM   #1
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I need help to prove

Given a convex subset K is a subset of R^3.

I(K)={z| there exists x, y in R such that (x,y,z)^t is in K} $\displaystyle \subseteq$ R.
I need to prove that I(K) is a convex subset. Can anybody help me? I don't know where to start.
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June 29th, 2016, 08:05 AM   #2
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I would think that a good place to start proving that a set was convex would be with the definition of "convex"! A set, X, is convex if and only if, for any two points, p and q, in X, every point "between" them, tp+ (1- t)q, for 0< t< 1, is also in X.

Here, suppose that $\displaystyle z_1$ and $\displaystyle z_2$ such "there exist $\displaystyle x_1$, $\displaystyle y_1$ and $\displaystyle x_2$ and $\displaystyle y_2$ such that $\displaystyle (x_1, y_1, z_1)$ is in K and $\displaystyle (x_2, y_2, z_2)$ is in K. What can we say about $\displaystyle tz_1+ (1-t)z_2$?

(Use the fact that K is convex: for any t between 0 and 1, $\displaystyle t(x_1, y_1, z_1)+ (1- t)(x_2, y_2, z_2)$ is in K.)
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