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We investigate the following question: what is the set of unit volume whichcan be best irrigated starting from a single source at the origin, in the senseof branched transport? We may formulate this question as a shape optimizationproblem and prove existence of solutions, which can be considered as a sort of"unit ball" for branched transport. We establish some elementary properties ofoptimizers and describe these optimal sets A as sublevel sets of a so-calledlandscape function which is now classical in branched transport. We prove$\beta$-H{\"o}lder regularity of the landscape function, allowing us to get anupper bound on the Minkowski dimension of the boundary: dim $\partial$A $\le$ d-- $\beta$ (where $\beta$ := d($\alpha$ -- (1 -- 1/d)) $\in$ (0, 1) is arelevant exponent in branched transport, associated with the exponent $\alpha$> 1 -- 1/d appearing in the cost). We are not able to prove the upper bound,but we conjecture that $\partial$A is of non-integer dimension d -- $\beta$.Finally, we make an attempt to compute numerically an optimal shape, using anadaptation of the phase-field approximation of branched transport introducedsome years ago by Oudet and the second author.

A fractal shape optimization problem in branched transport