I found one related isoperimetric inequality due to Schoenberg [1].
We say that a closed curve in $\mathbb{R}^{2n}$ is convex provided it never crosses a hyperplane more than $2n$ times.
Theorem. Let $\gamma$ be a closed convex curve in $\mathbb{R}^{2n}$ of length $L$. Let $K$ be the convex hull of $C$. Then $$ L^{2n}\geq (2\pi n)^n n! (2n)! |K|. $$
Moreover Schoenberg determined all cases of equality.
[1] Schoenberg, I. J. An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces. Acta Math. 91, (1954). 143-164.