Is this set of matrices convex? The set of positive definite matrices is convex. But what about this set?
$$\Omega = \left\{ (\mathbf{A}, \mathbf{b}) \in \mathbb{R}^{n \times n} \times \mathbb{R}^n : (\mathbf{A} - \mathbf{b}\mathbf{b}^\top) \text{ is p.d. } \right\}$$
I didn't have much luck trying to apply the definition.

Take $(A, b) \in \Omega$ and $(B, c) \in \Omega$ and let $\lambda \in [0, 1]$.
The question is whether $(C, d) := (\lambda A + (1-\lambda) B, \lambda b + (1-\lambda) c)$ is in $\Omega$, that is whether $C - d d^T$ is positive definite. But
\begin{align}
C - d d^T
&= \lambda A + (1 - \lambda) B - (\lambda b + (1 - \lambda) c) (\lambda b + (1 - \lambda) c)^\top \\
&= \lambda A + (1 - \lambda) B - (\lambda (b - c) + c) (b - (1 - \lambda) (b - c))^\top \\
&= \lambda A + (1 - \lambda) B - \lambda (b - c)b^\top - cb^\top + (1 - \lambda)c(b - c)^\top + \lambda (1 - \lambda) (b - c) (b - c)^T \\
&= \lambda (A - b b^T) + (1 - \lambda) (B - c c^T) + \lambda (1 - \lambda) (b - c) (b - c)^T,
\end{align} which is indeed positive definite.
