On the horseshoe conjecture for maximal distance minimizers

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality \mbox{max}_{y \in M} \mbox{dist}(y,\Sigma) \leq r for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$. Such sets can be considered shortest possible pipelines arriving at a distance at most $r$ to every point of $M$ which in this case is considered as the set of customers of the pipeline. We prove the conjecture of Miranda, Paolini and Stepanov about the set of minimizers for $M$ a circumference of radius $R>0$ for the case when $r < R/4.98$. Moreover we show that when $M$ is a boundary of a smooth convex set with minimal radius of curvature $R$, then every minimizer $\Sigma$ has similar structure for $r < R/5$. Additionaly we prove a similar statement for local minimizers.Comment: 25 pages, 21 figure

On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them

Independence numbers of Johnson-type graphs

We consider a family of distance graphs in $\mathbb{R}^n$ and find its independent numbers in some cases. Define graph $J_{\pm}(n,k,t)$ in the following way: the vertex set consists of all vectors from $\{-1,0,1\}^n$ with $k$ nonzero coordinates; edges connect the pairs of vertices with scalar product $t$. We find the independence number of $J_{\pm}(n,k,t)$ for $n > n_0 (k,t)$ in the cases $t = 0$ and $t = -1$; these cases for $k = 3$ are solved completely. Also the independence number is found for negative odd $t$ and $n > n_0 (k,t)$