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Consider subsets of $\mathbb{R}^n$. Instead of a straight line, every pair of points have to be joined by a line with the curvature of a parabola or less than it . This would include the straight line too. I guess one could arrive at some differentiability condition for the boundary curve in such cases. Is there something of this kind?

Edit: I just want the absolute value of curvature of the line joining it to be bound in some sense. I am not very particular about it being a parabola.

user2277550
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    Do you have an exemple of such a subset other than $\mathbb{R}^n$ itself ? – stity Mar 14 '18 at 16:33
  • @stity A cashewnut, half of a torus (roughly) , a region bounded by a smooth enough curve, I assume. I m not sure... – user2277550 Mar 14 '18 at 16:35
  • Can you state precisely what "a line with the curvature of a parabola or less than it" means? I can interpret that in a couple of ways. Pointwise curvature of a parabola can be arbitrarily large. The integral curvature is bounded. Parabola is also special for being a planar curve. –  Mar 14 '18 at 20:21
  • Given any ordered triple of distinct non-colinear points, there is a unique parabola passing through them in the given order. Thus given two points in your set, the only points in $\Bbb R^n$ that would not lie on some parabolic arc connecting them would be exactly the points on their line. But even then, if you take two points on a parallel line (which must be part of your set if it is "parabolically convex"), every point on the original line is on some parabolic arc through these two. So stity is right. The only such sets are $\Bbb R^n$ or sets with just 0 or 1 point. – Paul Sinclair Mar 15 '18 at 01:01
  • However, you can modify your concept a bit: a "parabolicly convex" set is such that given any three points in it, all points lying on a parabolic arc connecting two of the points and passing through the third must also be in the set. – Paul Sinclair Mar 15 '18 at 01:05
  • @PaulSinclair the graph of the function $y=x^2$ in $\Bbb R ^2$ does not satisfy OP's condition ? Although I'm not sure if I understand their definition ! – Red shoes Mar 15 '18 at 03:40

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