Related papers: The Mazur-Ulam theorem
We prove a simple inequality for a sum of squares of norms of two vectors in an inner product space. Next, using this inequality we derive the so--called "reverse uncertainty relation" and analyze its properties.
We will give a new proof for the Gromov's theorem on almost flat manifolds, which is an inductive proof on dimension.
In this paper, we introduce cone normed linear space, study the cone convergence with respect to cone norm. Finally, we prove the completeness of a finite dimensional cone normed linear space.
This is the first of two papers which aim to understand quasi-isometries of a subclass of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this…
A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in $\bR^2$. A more general theory is developed for frameworks in $\bR^3$ whose vertices are…
We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to a surjective real-linear isometry between the uniform algebras. It provides the first positive solution for Tingley's problem on a Banach…
We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.
We provide an algebraic description of the Teichm\"uller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may…
We present a geometric proof for the duality theorem of linear programming. Besides being self-contained and simple, the present approach also provides a transparent way for understanding this fundamental result.
We give a new proof of a theorem of Mansour and Sun by using number theory and Rothe's identity.
We prove an analog of the Szemer\'edi-Trotter theorem in the plane for definable curves and points in any o-minimal structure over an arbitrary real closed field $\mathrm{R}$. One new ingredient in the proof is an extension of the well…
We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last…
We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of…
It is proven a new analogue of the Theorem of Moser in a generalized context defined by Shilov Boundaries of Bounded and Symmetric Domains.
We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space…
We provide a simple and short proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints. The proof relies on an elementary linear algebra lemma and the local inverse theorem.
An technically interesting proof of a known theorem.
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric…
We study Borsuk-Ulam type results for the loopspace of an euclidean sphere without loops equal to their inverses.
A p-adic analogue of the pseudonorm version of the birational Torelli type theorem is obtained via a comparison theorem of image closures. Among other results obtained, we have a criterion for existence of rational points of canonically…