isepiphanic inequality

The classical isepiphanic inequality

$$36\pi V^2\leqq A^3,$$

concerns the volume $V$ and the area $A$ of any solid in ${ℝ}^3$.  It asserts that the ball has the greatest volume among the solids having a given area.

For a ball with radius $r$, we have

$$V=\frac{4}{3}\pi r^3,A=\mathrm{\hspace{0.33em}4}\pi r^2,$$

whence it follows the equality

$$36\pi V^2=A^3.$$

Cf. the isoperimetric inequality and the isoperimetric problem.

Title	isepiphanic inequality
Canonical name	IsepiphanicInequality
Date of creation	2013-03-22 19:19:08
Last modified on	2013-03-22 19:19:08
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51M25
Classification	msc 51M16