A spatial logic is any formal language with geometric interpretation. Research on region-based spatial logics, where variables are set to range over certain subsets of geometric space, have been investigated recently within the qualitative spatial reasoning paradigm in AI. We axiomatised the theory of (ROQ(R^2), conv, smaller or equal to) , where ROQ(R^2) is the set of regular open rational polygons of the real plane; conv is the convexity property and smaller or equal to is the inclusion relation. We proved soundness and completeness theorems. We also proved several expressiveness results. Additionally, we provide a historical and philosophical overview of the topic and present contemporary results relating to affine spatial logics.