For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$
such that images of every pairwise distinct $k$ points are affinely (linearly)
independent. Such maps are called affinely (linearly) $k$-regular embeddings.
We investigate the cohomology obstructions to existence of regular embeddings
and give some new lower bounds on the dimension $m$ as function of $X$ and $k$,
for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In
the latter case, some nonzero Stiefel-Whitney classes of $X$ help to improve
the bound.

In this paper a measure of non-convexity for a simple polygonal region in the
plane is introduced. It is proved that for "not far from convex" regions this
measure does not decrease under the Minkowski sum operation, and guarantees
that the Minkowski sum has no "holes".

We consider a continuous map $f :M\to N$ between two manifolds and try to
find some sufficient conditions for existence of self-coincidences, i.e. the
$q$-tuples of pairwise distinct points $x_1,..., x_q\in M$ such that $f(x_1) =
f(x_2) = ... = f(x_q)$.

We show that there are certain characteristic classes of vector bundle
$f^*TN-TM$ that guarantee the existence of self-coincidences for $f$. In
particular, we prove some non-trivial existence of self-coincidences for a
continuous map of a real projective space of certain dimension into a Euclidean
space.

In this paper the spaces of $q$-tuples of points in a Euclidean space,
without $k$-wise coincidences are studied (configuration-like spaces). A
transitive group action by permuting these points is considered, and some new
upper bounds on the genus (in the sense of Krasnosel'skii-Schwarz and
Clapp-Puppe) for this action are given. Some theorems of Cohen-Lusk type for
coincidence points of continuous maps to Euclidean spaces are deduced.

In this paper a version of Knaster-Kuratowski-Mazurkiewicz theorem for
products of simplices is formulated. Some corollaries for measure partition in
the plane and cutting families of sets in the plane by lines are given.