Journal of Convex Analysis 32 (2025), No. 4, 1145--1157
Copyright Heldermann Verlag 2025
Conditions for the Preservation of Motzkin Decomposability and the Pareto Bordered Property Under Addition
Bogdan Daniel Moldovan
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania
bogdan.moldovan1@ubbcluj.ro
Cornel Pintea
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania
cornel.pintea@ubbcluj.ro
We provide some sufficient conditions on pairs of Motzkin decompasable sets and Pareto bordered sets in order to get the Minkowski sum of their components Motzkin decomposable and Pareto bordered respectively. We also prove that minimal faces of a closed convex set are also extreme faces of the set and vice-versa. This result allows us to define the generalized Minkowski sets by using the extreme faces. A Minkowski type theorem is proved with extreme faces playing the role of the extreme points in the classical Minkowski Theorem. The special class of Pareto bordered sets, which is a subclass of that of generalized Minkowski sets, is also taken into account. Indeed, as mentioned above, we show that the Minkowski sum of some Pareto bordered sets with the same lineality remains Pareto bordered. Note that the class of generalized Minkowski sets is not closed with respect to the Minkowski sum. It is however worth to mention that the class of closed convex sets which are both Motzkin decomposable and generalized Minkowski (or shortly, MdgM sets) is closed both with respect to Minkowski sum and Cartesian product [see J. E. Martínez-Legaz and C. Pintea: Closed convex sets that are both Motzkin and generalized Minkowski sets, J. Nonlinear Var. Analysis 8/4 (2024) 571--579].
Keywords: Motzkin decomposable sets, Pareto bordered sets, generalized Minkowski sets.
MSC: 52A20, 53A07.
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