Inflated sets - Lycée Val de Durance (Pertuis)

Lycée Val de Durance (Pertuis)
We take a convex plane figure. The diameter of the figure is the greatest distance between two points of this figure. To this figure (A) we can "add" a point M outside the figure by considering the convex envelope formed by A and M. It is like placing a rubber band around A and M. A figure is said to be "inflated" when the addition of any point in the plane (in the above sense) increases its diameter.
Try inflating a square and other shapes. What can be said about inflated figures of the same diameter?
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