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Given a continuous closed curve $\gamma$ in $\mathbb R^n$ does there exist a convex body $K$ (convex set with non-empty interior) such that $\gamma\subset \partial K$?

I am looking for a reference to characterizations or special properties of these curves, or related results.

There is a discussion here, which define these curves, yet it proceeds in another direction. One way to characterize this class of curve would be to look at all curves such that $\gamma\subset\partial \langle\gamma\rangle$ and $\langle\gamma\rangle$ is a convex body (which would just mean that the curve is not contained in lower dimensional space), where $\partial$ denotes the boundary and $\langle\rangle$ denotes the convex hull. Another way would probably be existence of continuous choice of tangent planes such that curve lies on one side of it.

I am particularly interested in the characterization when curve $\gamma$ is piecewise linear and its a part of the boundary of a polytope which iterates through some vertices and some 1-dimensional facets.

Any references or ideas are welcome, thank you.

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    $\begingroup$ I don't know of any references, but in the piecewise linear case your second approach should yield an algorithm: Iterate through the line segments $\ell_i$ of $\gamma$, each of which is collapsed by a unique orthogonal projection $P_i$ to $\mathbb{R}^{n - 1}$. If for some segment $\ell_i$ the common projection of its endpoints lies in the interior of the convex hull of the projections of all other vertices of $\gamma$, then no such $K$ exists. Otherwise, it does. $\endgroup$
    – Tracy Hall
    Commented Oct 26, 2022 at 23:53
  • $\begingroup$ Will you clarify the sentence “I am looking for the reference to this characterization and related equivalent results”? Do you mean “I am looking for a reference to characterizations of these curves, or related results”? $\endgroup$
    – user44143
    Commented Oct 27, 2022 at 4:10
  • $\begingroup$ @MattF. Yes, sorry. In general, I don't know anything about this, and never thought about questions of this type. The standard convex geometry literature and the papers I am familiar with don't pursue this direction. This questions is vary vague on purpose because, in general I am interested in all possible results with similar perspective. Are there any special properties of the curves which lie on the boundary of convex bodies? What about properties of manifolds embedded in the boundary of convex bodies? A lot can be asked. $\endgroup$
    – Vadim Semenov
    Commented Oct 27, 2022 at 11:21
  • $\begingroup$ @TracyHall Seems very nice. I guess, this precisely means that $\gamma \subset\partial\langle\gamma\rangle$. And we can continue from there to construct a polytope. I also was interested if this polytope contains zero, but I guess your algorithm also answers this questions as well. It would be interesting to generalize this to continuous curves. I guess one will have to make a choice which projection to take at a point of a curve... Some section of tangent bundle (in a metaphorical sense, since strictly speaking it is not defined)... I will think about it. Thank you. $\endgroup$
    – Vadim Semenov
    Commented Oct 27, 2022 at 11:31

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