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.
Obviously you can check the variance of each attribute.

But unless the data is badly scaled, there will likely need the combination of attributes to explain the differences of clusters.