If I have $3$ convex or concave surfaces $E_0(x,y), E_1(x,y), E_2(x,y)$ that all intersect at $1$ and only $1$ point $(x^*, y^*)$, is it necessary that the function $$d(x,y)=\sum_{0\leq i<j\leq 2} (E_i(x,y)-E_j(x,y))^2$$ necessarily convex?
Obviously $d$ attains its minimum at $(x,y)=(x^*, y^*)$. But how would I go around showing it is convex?
Suppose $E_1(x,y) = x$, $E_2(x,y) = y^2$, $E_3(x,y) = 0$.
$d(x,y) = (x - y^2)^2 + x^2 + y^4$ which is not convex, because you can take two derivatives in $y$ and still have a $x$ term, which means the second derivative can be negative.