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In this note, we try to summarize the results we have so far in terms of the definition of the S-convexity phenomenon, but we also try to explain in detail the relevance of those. For some of those results, we dare presenting graphical illustrations to make our point clearer. S-convexity came to us through the work of Prof. Dr. Dragomir (2001) and Prof. Dr. Dragomir claimed to have had contact with the concept through the hands of Hudzik and Maligranda, who, in their turn, mention Breckner and Orlicz as an inspiration. We are working in a professional way with the phenomenon since the year of 2001, and that was when we presented our first talk on the topic. In that talk, we introduced a conjecture about the shape of S-convexity. We have examined possible examples, we have worked with the definition and examples, and we then concluded that we needed to refine the definition by much if we wanted to still call the phenomenon an extensional phenomenon in what regards Convexity. We are now working on the fourth paper about the shape of S-convexity and trying to get both limiting lines (negative and non-negative functions) to be as similar as possible. It is a delicate labour to the side of Real Analysis, Vector Algebra, and even Calculus.

SUMMARY AND IMPORTANCE OF THE RESULTS INVOLVING THE DEFINITION OF $S-$CONVEXITY