> ***Q***. Is there a higher-dimensional version of the theorem due to Fenchel that the
total curvature of a closed curve in $\mathbb{R}^3$ is $\ge 2\pi$,
with equality only if the curve is planar and convex?

Here the *total curvature* is 
$\int \kappa(s) \, ds$.
Fáry & Milnor subsequently proved that if the curve is knotted, then the total curvature
is strictly greater than $2\pi$. (Apologies if I have misinterpreted the history.)

By "higher-dimensional" I mean a result addressing the "total
curvature" (appropriately defined) 
of a closed, compact (hyper-)surface in $\mathbb{R}^d$, $d>3$.
In other words, I am seeking a generalization of the curve to a (hyper-)surface,
rather than a generalization of a 1D curve to $\mathbb{R}^d$, which I understand is addressed by Fáry/Milnor.

<hr />
> Milnor, John. "On total curvatures of closed space curves." *Mathematica Scandinavica* (1954): 289-296.
([Jstor link](http://www.jstor.org/stable/24488936), [eudml link](https://eudml.org/doc/165509).)