Consider a real algebraic set $A\subseteq\mathbb{R}^n$. The case I am interested in is a homogeneous cone (i.e., if $x\in A$ then also $\lambda x\in A$ for each scalar $\lambda>0$) where each stratum is contained in the closure of the higher-dimensional one, as in the case of the determinantal variety. Let $U\subseteq\mathbb{R}^n$ be a connected component of the complement $\mathbb{R}^n\setminus A$ such that $A$ is contained in the closure of $U$. Consider a path $\alpha$ in $A$ of finite length; say, a length-minimizing path for the induced metric. Can one always push $\alpha$ slightly into $U$ without increasing its length by more than $\epsilon$?
The relevant hypothesis here seems to be the local path-connectedness of the subset $A\subseteq\mathbb{R}^n$. I would appreciate a reference if possible particularly in the case of the real determinantal variety.