On properties of intersection of $\alpha$-sets

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 $\alpha$.