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.