How to prove that a concave production imply that the input requirement sets are convex? According to page 7 of this slide, &quot;A convex production set Y implies that the associated input requirement set V(y) is convex&quot;.
How can one go about proving it?
Start with definitions:


Production (possibilities) set: $Y$ which you know is convex
Input requirement set: $V(y)=\{\mathbf{x}:(y,−\mathbf{x})∈Y\}$


On page 7. you can see: $\mathbf{y}\in Y$ and $\mathbf{y'} \in Y$ which then implies $t\mathbf{y}+(1-t)\mathbf{y'} \in Y$. 

Hint 1:


   Remember that: $\mathbf{y}=(y,−\mathbf{x})$ !


What does it mean that $Y$ is a convex set? 


   $Y$ being a convex set means that if $\mathbf{y}=(y,−\mathbf{x}) \in Y$ and $\mathbf{y'}=(y,−\mathbf{x'}) \in Y$ then $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y$ as page 7. would suggest.


Okay, but what does that actually mean?


   $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y \implies (ty+(1-t)y, -t\mathbf{x}-(1-t)\mathbf{x'}) \in Y \implies (y,-t\mathbf{x}-(1-t)\mathbf{x'}) \in Y $


Which is just equivalent of saying that:


   $t\mathbf{x}+(1-t)\mathbf{x'} \in V(y)$ because both $\mathbf{x}$ and $\mathbf{x'}$ are in $V(y)$. (Look: second definition)