Abstract: We introduce the medial kernel, an association measure which provides for a robust construction of volume-aware distances defined directly on point clouds. The medial kernel is a similarity measure defined as the likelihood of two points belonging to a common interior medial ball. We use the medial kernel to construct a random walk on the point cloud, where movement in the walk is restricted to regions containing similar medial balls. Our distances are defined as the diffusion distances of this random walk, assigning low distance to points belonging to similar medial regions. These distances allow for a robust means of processing incomplete point clouds, capable of distinguishing nearby yet separate undersampled components, while also associating points which are far in Euclidean distance yet mutually share an interior volume. We leverage these distances for several applications: volumetric part segmentation, the construction of function bases, and reconstruction-by-parts - a surface reconstruction method which adheres to the medial kernel.