In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

In this paper, we will show that the spherical symmetric slices are the convex bodies that maximise the volume, the surface area and the integral of mean curvature when the minimum width and the circumradius are prescribed and the symmetric $2$-cap-bodies are the ones which minimise the volume, the surface area and the integral of mean curvature given the diameter and the inradius.

Let $X,Y$ be two Banach spaces and $B_{X}$ the closed unit ball of $X$. We prove that if there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists an isometry $F:X\rightarrow Y^{\ast \ast }$. If, in addition, $Y$ is weakly nearly strictly convex, then there is an isometry $F:X\rightarrow Y$. Making use of these results, we show that if $Y$ is weakly nearly strictly convex and there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists a linear isometry $S:X\rightarrow Y$.

We examine a class of ergodic transformations on a probability measure space $(X,{\it\mu})$ and show that they extend to representations of ${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.

Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset \cdots \,$. We prove that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ when $X$ is a Hilbert space.

In this paper, we establish a translation theorem for the generalised analytic Feynman integral of functionals that belong to the Banach algebra ${\mathcal{F}}(C_{a,b}[0,T])$.

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.

In this paper, we establish a convergence theorem for fixed points of generalised weak contractions in complete metric spaces under some new control conditions on the functions. An illustrative example of a generalised weak contraction is discussed to show how the new conditions extend known results.

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

We show that, in an inner product space $H$, the inequality

$$\begin{eqnarray}{\textstyle \frac{1}{2}}[\Vert x\Vert \,\Vert y\Vert +|\langle x,y\rangle |]\geq |\langle Px,y\rangle |\end{eqnarray}$$

$x,y$

$P:H\rightarrow H$