Advances on Ricceri's Most Famous Conjecture

New advances towards a (positive) solution to Ricceri's (most famous) Conjecture are presented. One of these advances consists of showing that a totally anti-proximinal absolutely convex subset of a vector space is linearly open. We also prove that if every totally anti-proximinal convex subset of a vector space is linearly open then Ricceri's Conjecture holds true. Finally we demonstrate that the concept of total anti-proximinality does not make sense in the scope of pseudo-normed spaces.