I think the Meissner tetrahedron has constant width, but without full tetrahedral symmetry. Let L be the longest side length of a scalene triangle. Then isn't the intersection of the 3 disks of radius L, centered at the triangle's vertices, a shape of constant width? --Dan Fred wrote:
Part of the problem is the misnomer "Reuleaux tetrahedron" attached to the figure you described, which is as you observe irrelevant to the discussion. The "Meissner" tetrahedron --- a special case of the construction I posed earlier --- may well have constant width, but my own previous attempts to decide this question have been inconclusive.
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So what does the constant-width superset (with the same diameter) of a scalene triangle (including interior) look like, anyway? And is it unique, given the triangle?