This manuscript develops a systematic theory of support functions and support-distance kernels. Starting from the pairwise distance geometry of function graphs, we introduce the basic notions of support families, anchored Gram kernels, and induced integral operators. We then establish their fundamental properties, including local expansions, evolution dynamics, variational principles, and spectral invariants. The framework is subsequently extended to abstract and generalized settings, where it provides a unified language for reconstruction, operator theory, and topological structures. Throughout the work, a careful distinction is maintained between proved results, conditional statements, and open problems. The outcome is a self-contained mathematical theory built on the concept of support-distance data, together with explicit conditions under which it may connect to broader problems in geometry, topology, and analysis.
Jianming Wang (Sat,) studied this question.