Generalizations and stability of convexity

Abstract

This dissertation consists of two distinct topics, separated into two main chapters. In Chapter 1, we introduce and investigate the algebraic structures called cornets. Our primary focus is on extending the well-regarded Rådström cancellation principle to these specific objects. This chapter aims to delve into the adaptation and application of the cancellation principle within the context of cornets. Additionally, we establish the fundamental properties of cornets, covering essential aspects such as convexity properties, topological notions and, boundedness. In Chapter 2, we broaden the concepts of convex, concave, and affine sequences by introducing the novel notions of q-convex, q-concave, and q-affine sequences. This chapter not only unveils foundational results but also highlights an unexpected correlation between these sequences and Chebyshev polynomials. Furthermore, we present two practical applications of q-convex, q-concave, and q-affine sequences. The first involves addressing a minimax-type problem, while the second is an application in fixed point theory.