# Find shared properties of a cluster samples

I have a dataset which contains ~15 features. With the elbow method, I found out that the optimal number of clusters is probably four. Therefore, I applied the K-means algorithm with four clusters. Now, I would like to understand why these clusters have been formed the way they are. In other words, I would like to know what are the shared properties of the points of a specific cluster.

My idea is the following:

Let's pretend that C1 are the coordinates of the centroid of the first cluster and that P1 and P2 are two points of this cluster.

$$C1 = \begin{pmatrix} 5\\ 2\\ 4\\ \end{pmatrix}$$

$$P1 = \begin{pmatrix} 8\\ 2\\ 6\\ \end{pmatrix} P2 = \begin{pmatrix} 9\\ 2\\ 0\\ \end{pmatrix}$$

If we compute the average distance of the different coordinates of P1 and P2 we obtain this:

$$DistAverage = \begin{pmatrix} ((8-5)+(9-5))/2\\ ((2-2)+(2-2))/2\\ ((6-4)+(4-0))/2\\ \end{pmatrix} = \begin{pmatrix} 3.5\\ 0\\ 3\\ \end{pmatrix}$$

Would this mean that the second feature is a "shared property" of the points of this cluster (since the average distance is 0) ?

I hope that the question was clear enough.