We present a theorem on the existence of local continuous homomorphic inverses of surjective Borel homomorphisms with countable kernels from Borel groups onto Polish groups. We also associate in a canonical way subgroups of $ℝ$ with certain analytic P-ideals of subsets of $\mathbb{N}$. These groups, with appropriate topologies, provide examples of Polish, nonlocally compact, totally disconnected groups for which global continuous homomorphic inverses exist in the situation described above. The method of producing these groups generalizes constructions of Stevens and Hjorth and, just as those constructions, yields examples of Polish groups which are totally disconnected and yet are generated by each neighborhood of the identity.

The main aim of this paper is to prove that every non-$\sigma$-lower porous Suslin set in a topologically complete metric space contains a closed non-$\sigma$-lower porous subset. In fact, we prove a general result of this type on “abstract porosities.” This general theorem is also applied to ball small sets in Hilbert spaces and to $\sigma$-cone-supported sets in separable Banach spaces.

In our previous work, we obtained sufficient conditions for the existence of trajectories with unbounded consumption for a model of economic dynamics with discrete innovations. In this paper, using the porosity notion, we show that for most models these conditions hold.

This paper contains a review of recent results concerning typical properties of dimensions of sets and dimensions of measures. In particular, we are interested in the Hausdorff dimension, box dimension, and packing dimension of sets and in the Hausdorff dimension, box dimension, correlation dimension, concentration dimension, and local dimension of measures.

We generalize Stechkin's theorem about the single-valuedness of the nearest point mapping at most points to any complete length space without bifurcating geodesics and make further observations.

The aim of this paper is to present some generic existence results for nearest and farthest points in connection with some geometric properties of Banach spaces.

We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $\sigma$-porous in corresponding spaces. Some applications to optimization are given.

In this paper, we will present some of the latest advances that have occurred in the study of weak Asplund spaces. In particular, we will give an example of a Gâteaux differentiability space that is not weak Asplund.

We study the minimization problem $f\left(x\right)\to min$, $x\in C$, where $f$ belongs to a complete metric space $ℳ$ of convex functions and the set $C$ is a countable intersection of a decreasing sequence of closed convex sets $C_i$ in a reflexive Banach space. Let $ℱ$ be the set of all $f\in ℳ$ for which the solutions of the minimization problem over the set $C_i$ converge strongly as $i\to \infty$ to the solution over the set $C$. In our recent work we show that the set $ℱ$ contains an everywhere dense $G_{\delta }$ subset of $ℳ$. In this paper, we show that the complement $ℳ\backslash ℱ$ is not only of the first Baire category but also a $\sigma$-porous set.