This article is cited in
1 paper
MATHEMATICS
On properties of intersection of $\alpha$-sets
A. A. Ershovab,
O. A. Kuvshinovb a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16,
Yekaterinburg, 620219, Russia
b Ural Federal University, ul. Mira, 19, Yekaterinburg, 620219, Russia
Abstract:
In this paper, we study the properties of
$\alpha$-sets, which are one of the generalizations of convex
sets. In the first part of the paper, the equivalence of two definitions of
$\alpha$-sets in the plane is
proved. The second part of the work is devoted to the experimental study of the properties of simply
connected intersections of
$\alpha$-sets. It follows from the results of numerical experiments that the
value
$\alpha$ of the measure of nonconvexity in a simply connected intersection of two
$\alpha$-sets
can be greater than the initial value of
$\alpha$ in intersected sets even when these values are very close
to zero. Based on these results, we can hypothesize that, firstly, such an increase in the value of
$\alpha$
is possible with an arbitrarily small initial
$\alpha$ for intersected sets, secondly, this increase is limited
by a linear function of the initial value of
$\alpha$.
Keywords:
generalized convex set, $\alpha$-set, intersection of sets.
UDC:
517.977
MSC: 52A01,
11H16 Received: 10.02.2020
DOI:
10.35634/2226-3594-2020-55-06