Abstract
The paper considers \(\alpha \)-sets that are the generalization of convex sets. This concept was introduced by V.N. Ushakov in the 2000s to classify the reachable sets of controlled systems according to the degree of their nonconvexity. Since then a lot of properties of such sets have been discovered and proven. However, not all the “natural” properties are fulfilled. We have proved two “unnatural” properties for such sets in the paper. Firstly, we provide an example of a non-self-intersecting curve, a connected segment of which is “less convex” than the entire curve in terms of \(\alpha \)-sets. Secondly, we show that there is an \(\alpha \)-curve which is not representable as a graph of the function for all \(\alpha >0\).
The reported study was funded by RFBR according to the research project no. 18-01-00018 mol_a. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0006.
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Ushakov, V., Ershov, A., Pershakov, M. (2019). Counterexamples in the Theory of \(\alpha \)-Sets. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_26
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DOI: https://doi.org/10.1007/978-3-030-33394-2_26
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