On the optimal shape of tree roots and branches
Living organisms come in an immense variety of shapes, such as roots, branches, leaves, and flowers in plants, or bones in animals.
In many cases it is expected that, through natural selection, these organisms have evolved into a "best possible" shape.
From a mathematical perspective, it is thus of interest to study functionals whose minimizers correspond to some of the many shapes found in the biological world.
As a step in this direction, we consider two functionals, and show how they can be used to describe the optimal configurations of roots and branches in a tree.
The first one, which we call the sunlight functional, models the total amount of sunlight captured by the leaves of a tree.
The second one, which we call the harvest functional, models the total amount of nutrients collected by the roots.
The above functionals will be combined with a "ramified transportation cost", for transporting nutrients from the roots to the base of the trunk,
or from the base of the trunk to the leaves.
The talk will address the semicontinuity of these functionals, and the existence and
properties of optimal solutions, in a space of bounded measures.