Some Properties of the Sum and Geometric Differences of Minkowski

Abstract

The sets of Minkowski algebraic sum and geometric difference are considered. The purpose of the research in this paper is to apply the properties of Minkowski sum and geometric difference to fractional differential games. This paper investigates the geometric properties of the Minkowski algebraic sum and the geometric difference of sets. Various examples are considered that calculate the geometric differences of sets. The results of the research are presented and proved as a theorem. At the end of the article, the results were applied to fractional differential games.

On the Geometric Properties of the Minkowski Operator

“…If we put equations ( 14) and ( 13) to relation (12), condition (6) is obtained. This completes the proof.…”

“…3) We check that the Minkowski difference of polygons P A and P B is not empty. For this, we calculate the expression (6) and check that if the relation ( 6) is fulfilled, the difference P A * P B is not empty, otherwise, the difference consists of an empty set; 4) Suppose that in relation (6)…”

“…Theorem 6.3. For polygons P A and P B in the Euclidean plane R 2 , condition (6) holds. If compact set N is completely dense embedded in set P B , then the equality P A * N = P A * P B holds.…”

“…Later, in other works such as [4,5], various properties of this "geometric difference" were studied, and with their help, the conditions for solving the problem of chasing were eased. Also, many geometric properties of Minkowski's difference and sum are presented in [6,7]. To date, several scientific researches have been conducted to find algorithms for calculating the Minkowski sum.…”