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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.

Mikhail Katz
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  • Can I ask for a clarification of "push $\alpha$ slightly into $U$"? I guess this means you also want the distance between $\alpha$ and the pushed curve $\alpha'$ to be less than some other given bound. (Otherwise one could just translate $\alpha$ into $U$ without changing its length.) – Lazzaro Campeotti Apr 19 '16 at 10:31
  • How would you do that? @potentially – Mikhail Katz Apr 19 '16 at 10:33
  • I am just talking about the case when $A$ is a cone. Then $U$ is also a cone, i.e preserved under multiplication by positive reals. In particular $U$ contains open balls of arbitrary radius, e.g. twice the diameter of $\alpha$. Translate $\alpha$ to a curve $\alpha'$ that passes through the centre of such a ball; then $\alpha$ is contained in $U$, and has the same length as $\alpha$. Am I understanding the question correctly? – Lazzaro Campeotti Apr 19 '16 at 10:38
  • No, here "homogeneous cone" means that if $x\in A$ then also $\lambda x\in A$ for each scalar $\lambda>0$. @potentially – Mikhail Katz Apr 19 '16 at 11:10
  • And the curve is supposed to be pushed out slightly, i.e., by an arbitrarily small amount, rather than moved to the center. @potentially – Mikhail Katz Apr 19 '16 at 11:16
  • Yes, that "slightly" is what I was asking for a specific description of in the question. (It seems to me that our definitions of homogeneous cone are the same.) – Lazzaro Campeotti Apr 19 '16 at 11:24
  • Right, I misunderstood your comment at first, sorry. – Mikhail Katz Apr 19 '16 at 11:39
  • I am a bit to lazy here to try to write a full proof, but my impression is that the answer is yes, probably for any semialgebraic set of dimension $<n$. Take a cellular decomposition of $\mathbf R^n$ adapted to $A$. This splits the curve into a finite number of arcs lying in a single cell. Move them successively one at a time avoiding the $A$ part of the cell... – ACL Apr 19 '16 at 16:55
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    Actually there are counterexamples for arbitrary algebraic sets. Take the union of the $z$-axis with the $xy$-plane. A segment $[-1,1]$ on the $z$-axis cannot be pushed out at all. One can easily mimick this to produce examples where it can be pushed out but the length is not controlled. @ACL – Mikhail Katz Apr 19 '16 at 16:57

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