Commonly complex cognitive concepts cannot consistently be connected to simple features of the world. Geometrical shape parameters and (e.g., edge features, compactness, color) may play a role for defining individual objects, but might be too variable to allow for concept formation. Earlier works had suggested that the formation of object concepts is strongly influenced by the division of our world along convex to concave surface transitions. In this first paper in a sequence of two we address this issue using abstract 3D geometrical structures (polycubes). In a first experiment, we let our subjects manipulate and compare polycubes with different compactness and different concavity/convexity asking which of them they would perceive as “an object.” Both parameters (compactness and concavity/convexity) are not correlated in these stimuli. Nonetheless, we find that subjects with clear prevalence choose compact and convex ones. We continue to ask how strongly this influences the way we construct objects. Thus, in a second experiment we let humans combine polycubes to form an object. Also here we find that they prefer compact and convex configurations. This suggests that this simple geometric feature may underlie our cognitive understanding of objectness not only with respect to perception but also by influencing how we build our world.