CONTENTSIntroduction.......................................................................................................... 5II. Quasi-transitive algebraic objects....................................................................... 12III. Rank of the quasi-transitivity of algebraic objects........................................... 22IV. Commutative algebraic objects.......................................................................... 25V. Regular algebraic objects........................................................................................
There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite-Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality which is valid for all convex functions is valid only for convex functions.
There are many types of midconvexities, for example Jensen convexity, t-convexity, (s,t)-convexity. We provide a uniform framework for all the above mentioned midconvexities by considering a generalized middle-point map on an abstract space X.
We show that we can define and study the basic convexity properties in this setting.
We provide a unified approach to different types of shadowing. This enables us to generalize some known shadowing result.
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