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I read here that the shortest distance between two differentiable non-intersecting curves is along their common normal.

But if we consider $y^2=x+1$, $y^2 = x$, their common normal is actually the largest possible distance:

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Maybe this is because the minimum distance between these two curves never exists in the first place... Is this a condition we should add in the statement?

Thanks

MangoPizza
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  • Does the last paragraph of the longer answer in the linked post address your question? – Elliot Yu Aug 15 '22 at 18:04
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    "This is only true of local minimums, so if the curves have endpoints then the absolute minimum might contain an endpoint on one or both of the curves. It is also true that not all common normals are even local minima, they could be local maxima, or they could even be saddle points where changing the position of the point on one curve increases the distance and changing it for the other curve decreases the distance, so the condition of a common normal is necessary but not sufficient for points on nonintersecting differentiable curves to be a local minimum of the distance." – Elliot Yu Aug 15 '22 at 18:04
  • Technically, an extremum (or inflection point) occurs at the location of a common normal. – David G. Stork Aug 15 '22 at 18:22

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