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 $C$ 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.  

The paper of Schoenberg has 12 citations in MathSciNet and some of these papers contain generalizations of Schoenberg's result. In particular Tilli [1] (Corollary 1.3) proved

> **Theorem.** Assume $C\subset\mathbb{R}^n$ is a compact set.  Let $K$ be the convex hull of $C$. Then $$ \mathcal{H}^1(C)^n\geq (n!)^2|K|,
$$ where $\mathcal{H}^1$ stands for the Hausdorff measure.

Clearly the result applies to curves in $\mathbb{R}^n$ of finite length.

**[1] Schoenberg, I. J.,**
An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces. 
*Acta Math.* 91, (1954). 143-164. 

**[2] Tilli, P.,**
Isoperimetric inequalities for convex hulls and related questions. 
*Trans. Amer. Math. Soc.* 362 (2010),  4497-4509.