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.