Isoperimetric Inequalities are inequalities concerning the area of a figure with a given perimeter. They were worked on extensively by Lagrange.
If a figure in a plane has area and perimeter then . This means that given a perimeter for a plane figure, the circle has the largest area. Conversely, of all plane figures with area , the circle has the least perimeter.
Note that due to this inequality, it is impossible to have a figure with infinite volume yet finite surface area.