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I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it.

Consider a finite family of hyperplanes in a finite-dimension real vector space $V$. What does it mean that a piecewise-linear path in $V$ is in general position with respect to the family of hyperplanes? And how do you prove that a piecewise-linear or smooth path can be approximated by one in general position?

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A linear segment would be considered "in general position" if it is not in a special position, that is one wishes to exclude things like

  • an end point of the segment is in any of the given hyperplanes
  • the direction of the segment is orthogonal to any of the given hyperplanes

The exact mileage may vary and depends on assumptions needed e.g. in a proof that follows. Then a piecewise-linear path in general position would be one with all linear segments in general position. The important thing is that any path must be changeable to one in general positione by arbitrarily small changes (the paths in general position should be an open and dense subset of all paths); for example if you have a path with one vertex in one of the hyperplanes, you can move that vertex by $\epsilon$ in the direction orthogonal to the hyperplane (and only finitely many choices will take you to another hyperplane). It is this last fact by which any piecewise-linear path can be approximated by one in general position (and hence so can smooth paths).

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