# Curves of constant width that contain triangles

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?

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

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.