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

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] H. Federer, W. H. Fleming, Normal and integral currents. Ann. of Math. 72 (1960), 458-520.

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. (MathSciNet review).

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

• I found Tilli's paper extremely interesting, thanks for pointing it out! Accidentally do you know if the Problem 1.3 (mentioned there in the paper) has a solution now? (I have posted the question here in a separate topic). Thanks again for the interesting reference! Feb 9, 2019 at 20:59
• @Romeo I am happy to see that my miserable activity on Mathoverflow sometimes makes some sense. Feb 10, 2019 at 21:54
• Don't even say it as a joke :-)! I do find your contributions really interesting and I thank you for your activity on MO. (Btw thanks also for the answer concerning estimates of incremental quotients in Sobolev spaces: I need some time to digest it :-)). Looking forward to reading you again soon ;-) Feb 10, 2019 at 22:06