$\bullet$ Every convex region of area $1$ contains a triangle of area at least as large as the area of the equilateral triangle inscribed in the circle of area $1$. Moreover, if the region does not contain a triangle of larger area, then it is an ellipse. [E. Sas, *Über eine Extremaleigenschaft der Ellipsen*, Compositio Math. 6 (1939), 468–470]. (Actually, Sas proves a more general statement, for $n$-gons with $n\ge3$.)

$\bullet$ Among all triangles of a given area the equilateral one, and no other, is of minimum perimeter.

$\bullet$ If the maximum area triangle inscribed in an ellipse is equilateral, then the ellipse is a circle.

Putting it all together:

**Theorem.** *If $C$ is a convex region with area $1$, then there exists a triangle contained in $C$ whose perimeter is at least as large as the maximum perimeter of a triangle inscribed in the circle of area $1$, and if $C$ does not contain a triangle of perimeter larger than that, then $C$ ***must be** a circle.

SIAM Journal on Computing, 14, no. 1 (1985): 134-147. $\endgroup$ – Joseph O'Rourke Mar 1 '14 at 0:271more comments