Consider the following function:
$$f(c_1,c_2) = \min((y-c_1)^2, (y-c_2)^2),$$
where $y$ is a fixed value, while $c_1, c_2$ are scalar arguments. The function is symmetric along $c_1 = c_2$ plane. The plane where $c_1 < c_2$ is the mirror image of the plane where $c_2 < c_1$. Both sub-planes are convex, obviously. What if $y$ is a two-dimensional vector and so are $c_1, c_2$? What are two sub-spaces which are mirror images of each other in this case? My guess is $||c_1||_2 < ||c_2||_2$ and $||c_2||_2 < ||c_1||_2$ will form two sub-spaces which are mirror images of each other. How could I prove or generalize it?