我们调查以下问题：什么是一套单位体积，可以最好地灌溉从一个单一的来源在原点，在这个意义上的分支运输？我们可以把这个问题表示为一个形状最优化问题，并证明其解的存在性，它可以看作是一种分支运输的“单位球”。我们建立了一些基本性质的优化和描述这些最佳集A作为子级集的所谓的景观功能，现在是经典的分支运输。我们证明了$β$-H {<$}lder正则性的景观功能，使我们能够得到一个上界的Minkowski维数的边界：dim $\partial$ A $\le$ d -- $β$（其中$β$：= d（$α$ --（1 - 1/d））$\in$（0，1）是一个相关的指数在分支运输，与指数$α$ > 1 -- 1/d出现在成本）。我们不能证明上界，但我们推测，$\partial$ A是非整数维d - $β$。最后，我们试图计算数值的最佳形状，使用自适应的相场近似的分支运输介绍了几年前由Oudet和第二作者。
英文摘要：
We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sense of branched transport? We may formulate this question as a shape optimization problem and prove existence of solutions, which can be considered as a sort of "unit ball" for branched transport. We establish some elementary properties of optimizers and describe these optimal sets A as sublevel sets of a so-called landscape function which is now classical in branched transport. We prove $β$-H{ö}lder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: dim $\partial$A $\le$ d -- $β$ (where $β$ := d($α$ -- (1 -- 1/d)) $\in$ (0, 1) is a relevant exponent in branched transport, associated with the exponent $α$ > 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 -- $β$. Finally, we make an attempt to compute numerically an optimal shape, using an adaptation of the phase-field approximation of branched transport introduced some years ago by Oudet and the second author.
