For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take $\dfrac{2}{\sqrt[4]{3}}$ as the length of each segment, so the unit triangle has area $1$.


It turns out that for small $n$, all the optimal solutions are *Steiner trees*, in the sense that each point belonging to more than one segment is the common endpoint of either two segments forming a straight line, or of three segments forming angles of $\dfrac{2\pi}{{3}}$. We'll call points of the latter kind *branching points*.


Using Steiner trees with only one branching point and the three legs of almost same length, we get the trivial lower bounds  

$ \ \ \,\quad A(3k)\geqslant 3k^2$,  

$A(3k+1)\geqslant k(3k+2)$,  

$A(3k+2)\geqslant (k+1)(3k+1)$.


Equally, we can write $A(n)\geqslant \left[\frac{n^2}3\right]= $ [A000212](http://oeis.org/A000212) $(n)$. The oeis entry contains several interesting comments but nothing which immediately applies here.


For a given $n$, there can be other Steiner trees of same length with more than one branching point which yield the same area. It might be an interesting question how many non-isomorphic ones exist, but before that, I want to ask the following:



> 
> * Is this bound sharp for all $n$?
> * Are there optimal solutions which are *not* Steiner trees?
> 
> 
> 


*(EDIT: in fact, for $n=2,3,4$ taking $n$ consecutive edges of a regular hexagon yields also optimal solutions, which I would however consider marginal.)*  

This is somewhat converse to the problem of [Steiner minimal trees for convex polygons](http://en.wikipedia.org/wiki/Steiner_tree_problem) but not exactly.


The problem, the above construction and both questions can be immediately generalized to trees spanning volumes in $\mathbb R^d$ instead of $\mathbb R^2$. Conjecturally the answers are the same as for the plane.



Trees with a maximal convex hull: are the only optimal solutions Steiner trees?
One may ask an analogous continuous problem:



> 
> Which connected set composed of simple arcs of total length $1$ has the largest convex hull?
> 
> 
> 


If my computations are correct, the star formed by three segments of length $1/3$ and forming the angles $2\pi/3$ gives a triangle of area $\frac{1}{4\sqrt{3}} \sim \frac{1}{6.93}$, whereas a half-circle of length $1$ gives a half-disc of area $\frac{1}{2\pi} \sim \frac{1}{6.28}$, which is larger.


This implies that for large enough $n$, the Steiner tree with one branching point is not optimal.




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**Edit:**
I just found that this problem has been solved in the special case that the connected set is a curve; see the following question:
[Largest convex hull of a unit length path](https://mathoverflow.net/questions/141651/largest-convex-hull-of-a-unit-length-path)


The three-dimensional case is discussed here:
[Largest possible volume of the convex hull of a curve of unit length](https://mathoverflow.net/questions/83026/largest-possible-volume-of-the-convex-hull-of-a-curve-of-unit-length)


