I have a seemingly easy problem about the proximal point in a variety of a general point in the ambient space, which I don't have a proof:
Let $X \subset \mathbb{C}^N$ be the affine cone over some nondegenerate projective variety $\tilde{X} \subset \mathbb{P}^{N-1}$, and $Z \subset X$ be a subvariety. For any $Q \in \mathbb{C}^N \setminus X$, there is some $Q_0 \in X$ such that $\|Q-Q_0 \| = \min_{p \in X} \|Q-p \|$, where $\| \cdot \|$ is a Euclidean distance. Can we conclude that for a general $Q$, the corresponding $Q_0 \notin Z$?
Thanks a lot for advice.