On sets minimizing their weighted length in uniformly convex separable Banach spaces (Q340447)

In this paper, a study of one dimensional geometric variational problems in an ambient Banach space. Existence and partial regularity issues are investigated. The paradigmatic Steiner problem is: minimize \(\int_C w d\mathcal{H}^1\) among compact connected sets \(C\subseteq X\) containing \(F\), where \(\mathcal{H}^1\) denotes the one dimensional Hausdorff measure (relative to the metric of \(X\)), \(w:X\to (0,+\infty)\) is a weight, and \(F\) is a finite set implementing the boundary condition. Assuming that the considered problem admits finite energy competing sets, the authors prove existence of a minimizer in case \(X\) is the dual of a separable Banach space, and \(w\) is weakly * lower semicontinuous and bounded away from zero. The authors allow for a varying weight \(w\) and obtain the lower semicontinuity of the weighted length. In studying the regularity of a minimizer \(C\) of the considered problem, the authors regard \(C\) as a member of a larger class of almost minimizing sets. In section 4 of the paper, the authors report on some properties of \((\xi,r_0)\) almost minimizing sets \(C\) in general Banach spaces. Sometimes it is assumed that the gauge \(\xi\) verifies a Dini growth condition. In section 5, the authors improve on the regularity of \(reg(C)\), by assuming that the ambient Banach space \(X\) is uniformly round. In section 6 the authors apply the obtained existence and regularity results to quasi-hyperbolic geodesics for instance in \(L_p\) spaces. Then, they apply the above results for the case where dim \(X\)=2, and the norm of \(X\) is round and \(C^2\).