Please help me with the following question.
Suppose that the constant level contours of some function $V:\mathbb{R}^{2} \rightarrow \mathbb{R}$ have the shape of a symmetric 6-arm star. Can such a function be convex?
Prove using convex definition $V (\alpha x_{1} + (1-\alpha) x_{2}) \leq \alpha V(x_{1})+(1-\alpha)V(x_{2}) $
The above proposition can be proved as follows. Let's define set
$$ C_{V} = \{(x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3} \;|\; x_{3} \geq V(x_{1}, x_{2})\} $$
It's convex by definition. Then any constant level of function V can be represented as a intersection of $C_{V}$ with some plane {z = const}, which is convex.