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The extremal problems, generously populating all branches of mathematics,
use only scalar targets. Problems with many objectives have become
the topic of research rather recently and noticeably beyond
mathematics, which explains the substantial gap between the levels of complexity
and power of the mathematical tools available for single objective and multiple
objective problems. This challenges the task of enriching the stock
of vector optimization problems within the theoretical core of mathematics.
Under study is the class
of geometrically meaningful vector optimization problems whose solutions
can be found explicitly to some extend in terms of conditions on surface area
measures. The positivy technique
of settling the extremal problems of convex geometry
is still insufficiently popular, while bridging the gaps between
mathematics and the art and science
of multiple criteria decision making.