I know that affine equalities such as
$$a_1 x_1 + a_2 x_2 + a_3 x_3 = b_3$$
are convex. I'm wondering whether there are nonlinear equality constraints, for instance something like
$$a_1 x_1 + a_2 x_2 + a_3 x_3 = b_3 + x^{2p},\quad p \in N$$
or even something like
$$x_1 = x_2^2$$
Are these kinds of equality constraint also convex? Can you give me any other example showing non-affine convex functions? Thanks!