# Derive the input requirement set from production set

This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like this question but my emphasize is different.

Q: True or False? If V(y) is a convex set, then the associated production set must be convex.

The input requirement set is defined as the set of all inputs required to produce at least the quantity $$y$$, or in other words, with a production function given by $$y=f(x),$$ $$V(y)= \{x: f(x) \geq y \}.$$ Note that $$x$$ can be a vector in this definition.
If $$f:\mathbb{R} \rightarrow \mathbb{R}$$ and is increasing, the input requirement is equivalently written as $$V(y)= \{x: x \geq f^{-1}(y) \}.$$ If $$f:\mathbb{R}^2 \rightarrow \mathbb{R}$$ and is increasing, the input requirement is equivalently written as $$V(y)= \{ ( x_1,x_2 ) : x_2 \geq f^{-1}(x_1,y) \}.$$
The minimum requirement on $$f$$ to obtain a convex set $$V$$, is that $$f$$ is quasi-convave in $$x$$. See for instance the Simon and Blume (1994, chapter 21.3) on this topic.