A d-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and p is a map from V to $\mathbb {R}^d$. The length of an edge $uv\in E$ in $(G,p)$ is the distance between $p(u)$ and $p(v)$. The framework is said to be globally rigid in $\mathbb {R}^d$ if the graph G and its edge lengths uniquely determine $(G,p)$, up to congruence. A graph G is called globally rigid in $\mathbb {R}^d$ if every d-dimensional generic framework $(G,p)$ is globally rigid.

In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graph G is strongly reconstructible in $\mathbb {C}^d$ if the set of (unlabeled) edge lengths of any generic framework $(G,p)$ in d-space, along with the number of vertices of G, uniquely determine both G and the association between the edges of G and the set of edge lengths. It is known that if G is globally rigid in $\mathbb {R}^d$ on at least $d+2$ vertices, then it is strongly reconstructible in $\mathbb {C}^d$. We strengthen this result and show that, under the same conditions, G is in fact fully reconstructible in $\mathbb {C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstruct G, without any constraint on the number of vertices (although still under the assumption that the edge lengths come from a generic realization).

As a key step in our proof, we also prove that if G is globally rigid in $\mathbb {R}^d$ on at least $d+2$ vertices, then the d-dimensional generic rigidity matroid of G is connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^{d}$ for some $n$ and some $d\geqslant 2$. Each pair of points has a Euclidean distance in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair distances corresponding to the edges of $G$. In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of $d$ and $n$. In this setting the distances are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point pair gave rise to which distance, nor is data about $G$ given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair distances (together with $d$ and $n$) if and only if it is determined by the labeled distances.

In this paper, we prove that on a compact, $n$-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures ${{\mu }_{1}},\ldots ,{{\mu }_{m}}$ is absolutely continuous with respect to the $n$-dimensional Hausdorff measure if one of them is.

This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n-G$-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n-G$-quasi preserving property, and two kinds of $n$-isometries on $n-G$-quasi normed space are equivalent; we generalize the Benz theorem to $n$-normed spaces with no restrictions on the dimension of spaces.

Motivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.