Let A be a convex compact set in the plane (with a piecewise smooth boundary, say). We want to `inflate' it in such a way that the diameter does not increase.
More accurately, we are looking for all sets C such that
a) A is a subset of C; b) diam(A)=diam(C)
Let now B is the largest possible set C which satisfies these two properties.
By `largest' I mean either that it m(B) = max m(C), where m is the Lebesgue measure; or that B actually contains any C with these properties. Let us call B the isodiametric hull of A.
The simplest example of A is of course the square: here B is the superscribed disc, and it is the isodiametric hull of A in the strong sense.
Another example is the equilateral triangle, for which B is the Reuleaux triangle. Similarly, for any regular 2n-gon we have the disc, and for any regular (2n+1)-gon its isodiametric hull is a Reuleaux polygon.
The first non-trivial example that comes to mind is an isosceles triangle that isn't equilateral. It is clear that the hull is always a set of constant diameter but how does one actually obtain it? It seems that its boundary - a curve of constant width - is not a finite union of circular arcs.
I wonder if all this is well known (being such a natural question!). In particular, does the isodiametric hull of a set always exists in the strong sense?
Added: of course, if there is no IDH in the strong sense, B may not be unique. Its area is unique, though. How does one find it?