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.

See also: https://mathoverflow.net/q/322846/121665.

**[1] Schoenberg, I. J.,**
<A HREF="https://projecteuclid.org/euclid.acta/1485892064"><FONT FACE="Arial">An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces
</FONT></A><FONT FACE="Arial">.
*Acta Math.* 91, (1954). 143-164. 
<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=156700&sort=Newest&vfpref=html&r=127&mx-pid=65944%0A"><FONT FACE="Arial">(MathSciNet review)</FONT></A><FONT FACE="Arial">.




**[2] Tilli, P.,**
<A HREF="http://www.ams.org/journals/tran/2010-362-09/S0002-9947-10-04734-3/home.html"><FONT FACE="Arial">Isoperimetric inequalities for convex hulls and related questions. </FONT></A><FONT FACE="Arial">.
*Trans. Amer. Math. Soc.* 362 (2010),  4497-4509. 
<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=610927&sort=Newest&vfpref=html&r=19&mx-pid=2645038"><FONT FACE="Arial">(MathSciNet review)</FONT></A><FONT FACE="Arial">.