We show that any mapping between two real $p$ -normed spaces, which preserves the unit distance and the midpoint of segments with distance $2^{p}$ , is an isometry. Making use of it, we provide an alternative proof of some known results on the Aleksandrov question in normed spaces and also generalise these known results to $p$ -normed spaces.

In this paper we provide an elementary proof of James’ characterisation of weak compactness for Banach spaces whose dual ball is weak∗ sequentially compact.

We prove a DDVV inequality for submanifolds of warped products of the form $I\times _{a}\mathbb{M}^{n}(c)$ , where $I$ is an interval and $\mathbb{M}^{n}(c)$ is a real space form of curvature $c$ . As an application, we give a rigidity result for submanifolds of $\mathbb{R}\times _{e^{\unicode[STIX]{x1D706}t}}\mathbb{H}^{n}(c)$ .

Wu [‘An order characterization of commutativity for $C^{\ast }$ -algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$ -algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$ -algebras in terms of the order.

Motivated by the local theory of Banach spaces, we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalised roundness of metric spaces. We illustrate this technique by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if ${\mathcal{U}}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{\ast \ast }$ and the ultrapower $(X)_{{\mathcal{U}}}$ have the same generalised roundness as $X$ , and (2) no Banach space of positive generalised roundness is uniformly homeomorphic to $c_{0}$ or $\ell _{p}$ , $2<p<\infty$ . For metric trees, we give the first examples of metric trees of generalised roundness one that have finite diameter. In addition, we show that metric trees of generalised roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.

Let $\unicode[STIX]{x1D6E4}$ be a countable discrete group that acts on a unital $C^{\ast }$ -algebra $A$ through an action $\unicode[STIX]{x1D6FC}$ . If $A$ has a faithful $\unicode[STIX]{x1D6FC}$ -invariant tracial state $\unicode[STIX]{x1D70F}$ , then $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}\circ {\mathcal{E}}$ is a faithful tracial state of $A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}$ where ${\mathcal{E}}:A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}\rightarrow A$ is the canonical faithful conditional expectation. We show that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property if and only if both $(A,\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6E4}$ have the Haagerup property. As a consequence, suppose that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property where $\unicode[STIX]{x1D6E4}$ has property $T$ and $A$ has strong property $T$ . Then $\unicode[STIX]{x1D6E4}$ is finite and $A$ is residually finite-dimensional.

We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$ -space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$ . We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$ . The proof of our results relies on the structure topology.

We propose a new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. The method combines the Rivaie–Mustafa–Ismail–Leong conjugate gradient method for unconstrained optimisation problems and a new nonmonotone line-search method. The global convergence of the proposed method is established under some mild assumptions. Numerical results using 104 test problems from the CUTEst test problem library show that the proposed method is promising.

In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

In this paper we first give a negative answer to a question of Amini-Harandi [‘Best proximity point theorems for cyclic strongly quasi-contraction mappings’, J. Global Optim.56 (2013), 1667–1674] on a best proximity point theorem for cyclic quasi-contraction maps. Then we prove some new results on best proximity point theorems that show that results of Amini-Harandi for cyclic strongly quasi-contractions are true under weaker assumptions.

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$ , where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$ . A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established.

In this paper we prove Caristi’s fixed point theorem using only purely metric techniques.

A lattice walk with all steps having the same length $d$ is called a $d$ -walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$ -walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$ , we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$ . To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.