Geometry of Curves and Surfaces — Problem 5.1

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A curve is a continuous map $\gamma:[a,b]\to\mathbb R^3$. Its length is \[L(\gamma)=\sup_{a=t_0<\cdots<t_m=b}\sum_{j=1}^m\lVert\gamma(t_j)-\gamma(t_{j-1})\rVert.\] It is rectifiable if this is finite, and closed if $\gamma(a)=\gamma(b)$. Its width is \[w(\gamma)=\min_{u\in S^2}\left(\max_{t\in[a,b]}\langle\gamma(t),u\rangle-\min_{t\in[a,b]}\langle\gamma(t),u\rangle\right),\] the minimum distance between parallel planes enclosing it. Write $\operatorname{conv}(\gamma)$ for the convex hull of its image. Its inradius is \[r(\gamma)=\sup\{r>0:\text{some }c+rS^2\subset\operatorname{conv}(\gamma)\text{ is disjoint from }\gamma([a,b])\}.\] Thus the source imposes disjointness from the curve in addition to containment of the sphere in its convex hull. Consider positive prescribed width or inradius.

Statement recorded in the supplied source.

Determine the shortest rectifiable curve in $\mathbb R^3$ with a prescribed width, and determine the shortest such curve with a prescribed inradius. In each problem determine also the shortest curve when it is required to be closed. [S2]

Short English statement

Find the shortest space curve of a given width or inradius, and solve both problems also for closed curves.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7000020 (AMR-069-0020). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mohammad Ghomi, The length, width, and inradius of space curves, arXiv:1605.01144v3 (2018), Introduction, pp. 2--4; Section 2, definition of curve and length, p. 4. https://arxiv.org/pdf/1605.01144v3.