Remark 13.3.4 (Isoparametric Problems).label If $A\suf\R^{n}$ compact convex then
\begin{align*}\frac{V_{n-1}(A)^{n}}{V_{n}(A)^{n-1}}\geq \frac{V_{n-1}(B+2^{n})^{n}}{V_{n}(B_{2}^{n})^{n-1}}\end{align*}
with equality if and only if $A=B_{2}^{n}$. In $n=2$ for $\gam$ a smooth closed curve enclosing the domain $D$, this reduces to $P(\gam)^{2}\geq 4\pi Area(D)$.
Theorem 13.3.3 shows that Steiner symmetrization exists for the class of nonempty compact convex sets in $\R^{n}$.
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