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).