Question: A region in the plane is called convex if the line segment joining any two points in the region lies wholly inside the region. In symbols, $R$ is convex if, for all $(x_1,y_1)$ and $(x_2,y_2)$ in $R$, $(\lambda x_1 + (1-\lambda)x_2,\lambda y_1 + (1-\lambda)y_2) \in R$ for all $\lambda \in [0,1]$.
a) Prove that if $R$ and $S$ are convex, then $R \cap S$ is convex. (No problem with this part)
b) If $R$ and $S$ are convex, is $R \cup S$ always convex? Prove your answer. (No problem with this part)
c) Prove that if $R$ is convex, then the reflection of $R$ in the $x$-axis is convex. (No problem with this part)
d) If $R$ is convex, is the set,
$2R = \{(x,y):(x,y) = (2x',2y')$ for some $(x',y') \in R\}$
always convex? Prove your answer and illustrate with a diagram.
I've been successful in all parts except (d). I am not sure exactly what to do nor what this actually means. Any explanation or guidance is greatly appreciated!
Best Answer
If $(x,y)=(2x',2y'), (u,v)=(2u',2v')\in 2R ,$ then $$t(x,y)+(1-t)(u,v)=2 (t(x',y')+(1-t)(u',v'))\in 2R $$ since $R$ is convex.