Isoperimetric inequality for closed curves in $\mathbb{R}^n$

Question

A well known isoperimetric inequality for closed curves in $\mathbb{R}^2$ can be generalized to closed curves in $\mathbb{R}^{2n}$, see: https://mathoverflow.net/a/321505/121665.

I have two questions:

Question 1. Is it possible to prove a reasonable isoperimetric inquality for closed curves in $\mathbb{R}^{2n+1}$?

Questions 2. Can an inequality from https://mathoverflow.net/a/321505/121665 be generalized to smooth mappings of $\mathbb{S}^k$ to $\mathbb{R}^n$ for some $n>k+1$?

Has anyone seen any related results?

There is a general isoperimetric inequality for currents (Theorem 6.1 in [1]) that has been generalized in many ways, but I am looking for a more elementary statements, more in the spirit of https://mathoverflow.net/a/321505/121665.

[1] Federer, H.; Fleming, W. H., Normal and integral currents, Ann. Math. (2) 72, 458-520 (1960). ZBL0187.31301.

Answer 1

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: Maximizing an integral w.r.t. a measure on the unit sphere.

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

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