Optimal wireframe sphere

Question

Suppose you have a length $L$ of metal pipe at your disposal, and you would like to build a wireframe unit-radius sphere, by bending, cutting, and welding the pipe into a connected structure $F$. Your goal is to minimize the height difference between where the center of the unit sphere would be ($1$) and where the center of your structure $F$ is, in any position resting on flat ground.

To be more precise, let $S$ be a unit radius sphere with center $o$, and $F$ a connected structure, built from pieces that are topological arcs, inscribed in contained within $S$. Let $H$ be the convex hull of $F$, required to enclose $o$, and define $\delta$ to be $1$ minus the minimum distance from $o$ to any point on $H$. So $\delta$ is the difference between the spheres centered on $o$ circumscribed about and inscribed in $H$.

Q1. For a given $L$, what is $\delta_{\min}(L)$, the minimum value of $\delta$ for any connected structure $F$? Equivalently, given $\delta$, what is $L_{\min}(\delta)$, the minimum length of all pieces in $F$ together that achieve $\delta$?

For example, given $L=4 \pi$, one might construct two orthogonal hoops:
         
This achieves $\delta = 1 - \sqrt{2}/2 = 0.29$. But surely this is not optimal. For example, one could remove some pipe near the poles and add it to the equator to lower $\delta$. This suggests this question:

Q2. Is an optimal frame structure $F$ always composed of straight segments? I.e., does it ever help to bend the pipe?

I think not.

The question can be generalized to any dimension. Even in $\mathbb{R}^2$ it seems not uninteresting:
         

Q3. What are the optimal structures $F$ inscribed in a unit circle that minimize $\delta$ for a given $L$? [Added:] Could it be that a regular $n$-gon minus one edge is optimal for that $L$, as suggested by the hexagon example above?

This two-dimensional version especially feels like it should have been investigated previously, but I am not finding any literature on it. Ideas and/or pointers welcomed—Thanks!

Addendum. Thanks to Gerhard Paseman and Aaron Meyerowitz, it is now clear that for the 2D question Q3, the Steiner tree spanning the vertices of a regular $n$-gon is a strong candidate for optimality, and certainly improves upon the $n$-gon minus an edge (except for $n=6$, as per Aaron's remark).
   

Answer 1

For Question 3, in the case that $L=3\sqrt{2} \approx 4.24$ one can do better than 3 sides of a square. The two diagonals have length 4 with the extra left for improving thigs a bit. Probably a Steiner tree would be even more efficient (I think that your 5/6 of a hexagon example with all the 120 degree angles lucks into being a Steiner tree itself.)

LATER

Here is a structure F with $L=5$ and $\delta=0.08897$ I think this construction gives the minimum $\delta$ at least for certain $L.$ It certainly is an improvement over 5/6 of a hexagon and works for any $L.$

The illustration shows the unit circle and a concentric circle whose radius happens to be $r=0.9110275245$ along with three tangents. The straight segments run from $[2r^2-1,\pm2r\sqrt{1-r^2}]$ to $[r,\pm\sqrt{1-r^2}]$.

For $L \le \frac{\pi+2}{\sqrt{2}} =3.635655$ We achieve $r=\frac{L}{\pi+2} $ using the (left) half of a circle of radius $r$ along with two horizontal segments. The Steiner tree competition would presumably achieve $r=\frac{L}{6}$ using three segments of length$\frac{L}3$ to get convex hull an equilateral triangle.

Answer 2

It is unlikely that one can solve that type of problem. But one can consider this as a competition; who get a better upper/lower bound. For lower bound there is a nice construction which I learn from Zalgaller's paper.

Take regular $n$-gon in $xy$-plane with center at the origin and a side parallel to $y$-axis. Consider the circles formed by revolution of its vertices around $x$-axis in $xyz$-space. Cut the circles along $xy$-plane and leave only the upper part. Repeat the construction for the part below $xy$-plane, but for the line obtained by rotation of $x$-axis by angle $\tfrac\pi{n}$. You get a spiral-like curve.

I am sure this gives a better estimate than your example. One can do bit better but not much better.

Answer 3

Guessing that this is the construction (p.90) to which ε-δ refers. (I cannot read Russian):