Exercise 3.10: Brunn–Minkowski and Classical Isoperimetric Inequality
This exercise proves that the following versions of the Brunn-Minkowski inequality are equivalent: \begin{align} \label{eq:bm1}\tag{BM1} \vol(A + B)^{1/n} &\geq \vol(A)^{1/n} + \vol(B)^{1/n} \newline \label{eq:bm2} \tag{BM2} \vol(\lambda A + (1 - \lambda) B)^{1 / n} &\geq \lambda \vol(A)^{1/n} + (1 - \lambda) \vol(B)^{1/n} \newline \label{eq:bm3} \tag{BM3} \vol(\lambda A + (1 - \lambda)B) &\geq \vol(A)^{\lambda} \vol(B)^{1 - \lambda} \, , \end{align} where \(\lambda \in [0, 1]\) and \(A , B\) are (closed compact) convex bodies in \(\R^n\).
Throughout we will be using the following identity which holds for any \(c \geq 0\) and \(A \in \R^n\) \begin{align} \vol(c A) = c^n \vol(A) \, . \end{align}
(a): \(~\eqref{eq:bm1} \iff \eqref{eq:bm2}\)
For the \(\eqref{eq:bm1} \implies \eqref{eq:bm2}\) direction \begin{align} \vol(\lambda A + (1 - \lambda) B)^{1 / n} &\geq \vol(\lambda A)^{1/n} + \vol((1 - \lambda) B)^{1/n} \newline &= \lambda \vol(A)^{1/n} + (1 - \lambda) \vol(B)^{1/n} \, . \end{align} For the reverse direction \begin{align} \vol(A + B)^{1/n} = 2\vol\left(\tfrac{1}{2} A + \tfrac{1}{2} B\right)^{1/n} \geq \vol(A)^{1/n} + \vol(B)^{1/n} \, . \end{align}
(b): \(~\eqref{eq:bm2} \implies \eqref{eq:bm3}\)
By the weighted AM-GM inequality (or Jensen’s inequality) \begin{align} \lambda \vol(A)^{1/n} + (1 - \lambda) \vol(B)^{1/n} \geq \vol(A)^{\lambda / n} \vol(B)^{(1 - \lambda) / n} \, . \end{align}
(c): \(~\eqref{eq:bm3} \implies \eqref{eq:bm2}\)
If either \(\vol(A)\) or \(\vol(B)\) equals zero, the claim is trivial. Otherwise define \(C = \tfrac{A}{\vol(A)^{1/n}}\), \(D = \tfrac{B}{\vol(B)^{1/n}}\), and \begin{align} \delta = \frac{ \lambda \vol(A)^{1/n} }{ \lambda \vol(A)^{1/n} + (1 - \lambda) \vol(B)^{1/n} } \, . \end{align} Observing \(\vol(C) = \vol(D) = 1\), \eqref{eq:bm3} implies \begin{align} \vol\left( \delta C + (1 - \delta) D \right)^{1 / n} = \vol \left( \frac{ \lambda A + (1 - \lambda) B }{ \lambda \vol(A)^{1/n} + (1 - \lambda) \vol(B)^{1/n} } \right)^{1/n} \geq 1 \, . \end{align}