# 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 ) 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.

 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 .

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  (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.

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

 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