# Pyramid Match Kernel: How to fit grid around data points?

I am planning to use Pyramid Match Kernel method for object recognition from depth images: I am going to extract a set of feature vectors $x \in \mathbb{R}^d$ for each object instance and then I want to use the Pyramid Match Kernel as described in http://jmlr.csail.mit.edu/papers/volume8/grauman07a/grauman07a.pdf in Grauman's paper, in order to use the set of feature vectors for each object with a SVM classifier.

I have a problem to understand how exactly we are supposed to build bins around the data points to begin with. The paper gives the following definition:

What does "inter-vector" distance mean here? I think of the following: We consider all $x_i$ values for each $X$ in $S$. Then we find two vectors with smallest distance $d$ ,(unique ones) and then scale all vectors with $1/d$. Then $D$ is the maximum absolute value of a vector element in $S$.

Is this what the authors meant here? What about if we are going to test a new vector $y$, how should we scale it?