Alex gave a good answer, but I would just like to highlight a subtle problem with your claim about polynomial time solvability of SDPs. This depends on having inner and outer bounding balls to the feasible sets as part of the input. As Alex showed, without the inner ball it can be hard to tell whether the feasible set is infeasible or just extremely small.
Similarly, without the outer bounding ball it can be hard to tell whether the problem is infeasible or only admits extremely large feasible solutions. An example is the problem: \[ \text{minimize}\quad z_n \] \[ \text{subject to}\quad z_0 \geq 1, \quad \begin{bmatrix}1 & z_i \\\ z_i & z_{i+1}\end{bmatrix}\succeq 0. \] The optimal solution has $z_0 = 1$ and $z_{i+1} = z_i^2$ for all $i$, so $z_n = 2^{2^n}$. Even writing down the optimal objective value takes exponentially many bits in binary. It is conceivable that there is some other more compact but still useful encoding scheme (clearly there is for this particular problem, but this is just a toy example). However, I am not aware of any reasonable proposals for such an encoding.