arXiv subjects

Mehmet Kiliç

Publications and source records attributed to Mehmet Kiliç.

3 recordsLinked to original sources

arXiv2020

Isometric Embeddings of Finite Metric Trees into $(\mathbb{R}^n,d_{1})$ and $(\mathbb{R}^n,d_{\infty})$

We investigate isometric embeddings of finite metric trees into $(\mathbb{R}^n,d_{1})$ and $( \mathbb{R}^n, d_{\infty})$. We prove that a finite metric tree can be isometrically embedded into $(\mathbb{R}^n,d_{1})$ if and only if the number of its leaves is at most $2n$. We show that a finite star tree with at most $2^n$ leaves can be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$ and a finite metric tree with more than $2^n$ leaves cannot be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$. We conjecture that an arbitrary finite metric tree with at most $2^n$ leaves can be isometrically embedded into $(\mathbb{R}^{n}, d_{\infty})$.

math.MG↗

arXiv2015

Tight Span of Path Connected Subsets of the Manhattan Plane

We show that the tight span of a path connected subset of the Manhattan plane can be constructed in a very simple way by "hatching" the subset in horizontal and vertical directions and then taking the closure of the resulting set.

math.MG↗

arXiv2015

Tight Span of Subsets of The Plane With The Maximum Metric

We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any nonempty $X\subseteq\mathbb{R}^2_{\infty}$, a closed, geodesically convex and minimal subset $Y\subseteq\mathbb{R}^2_{\infty}$ containing $X$ is isometric to the tight span $T(X)$ of $X$.

math.MG↗