The Victor–Frog Function: an overlap–based metric on a quotient of volumetric shapes

Many everyday solids can be continuously deformed into one another, yet the intuitive “amount of effort” required depends on how volume is distributed and how sharp or rounded the shapes are. Motivated by informal clay-model reasoning, this paper introduces the Victor–Frog Function (VFF3D), a simple volumetric, target-relative similarity measure between three-dimensional shapes. After a canonical volume normalization, VFF3D is defined as a Jaccard-type similarity between measurable subsets of ℝ³ modulo null sets. We formalize the construction on a natural quotient space of shapes, prove that the associated distance is a genuine metric, and show that its induced topology coincides with convergence in measure of normalized shapes. Basic invariance properties under rotations and scaling are established, along with a higher-dimensional extension. A small voxel-based numerical experiment illustrates the behavior of the metric by comparing an egg-like shape, a cube, and a pyramid against a sphere. The goal is not to compete with sophisticated shape analysis frameworks, but to isolate a minimal, mathematically clean formulation of an intuitive notion of volumetric similarity.