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Wikipedia references: Curve of constant width, Reuleaux polygon.

We record a pair of questions on the same lines as Smallest 3-ellipses that contain triangles.

Questions:

  • How does one find and characterize the smallest curves of constant width that contain a given triangle? 'Smallest' can mean 'least area' or 'least perimeter' or... and may have different answers. We consider only convex constant width curves.

  • And what about the largest constant width curves that are contained in a given triangle?

Guess: The largest area constant width curve contained in any triangle might be a (possibly irregular) Reuleux polygon. The contained constant width curve with largest perimeter may not be formed of circular arcs.

Note added on October 18th, 2023: We can also ask inside-out versions of these questions; for example, given a constant width curve, is the triangle for which it is the smallest containing/ largest embedded constant width curve unique? How to find it/them?

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  • $\begingroup$ Thanks. Took off both the conics tag and the mention of higher dimensions! $\endgroup$ Commented Sep 8, 2021 at 16:36

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About the smallest curve of constant width, the longest edge, say $d$, of the given triangle, must serve as one of the diameters of the curve. The perimeter of such curves is $\ \pi\cdot d,\ $ and that’s the minimum for the requested curves.

When the largest angle of the given triangle is less or equal $\ \frac\pi2\ $ (the obtuse case), then the curve can be simply a circle.

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    $\begingroup$ I think you mean "by Barbier's theorem, the perimeter of any constant width curve of diameter d will be pi X d. So, if d is the length of the longest side of a triangle, we have a range of least perimeter constant width curves all with width d and perimeter pi X d and containing the triangle". So, we do have a whole range of equally correct answers for one of the questions - which is the constant width curve of smallest perimeter that contains a given triangle? Thanks. $\endgroup$ Commented Sep 9, 2021 at 8:12
  • $\begingroup$ @NandakumarR, somehow (strange) I replaced word perimeter by area. I'll fix it. $\endgroup$
    – Wlod AA
    Commented Sep 9, 2021 at 8:45
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    $\begingroup$ Thanks! And your answer also seems to yield that the Reuleux triangle with width/diameter equal to longest side of the triangle is the least area containing constant width curve. $\endgroup$ Commented Sep 9, 2021 at 9:28
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    $\begingroup$ As per en.wikipedia.org/wiki/Barbier%27s_theorem, Barbier's theorem does not generalize naturally to 3d. So, in 3d, the question of the least surface area surface of constant width that contains a tetrahedron might have as unique answer the surface of revolution of a Reuleux triangle. $\endgroup$ Commented Sep 9, 2021 at 10:08

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