Data discrimination after clustering

My task consists of two points:

1) Make data clustering;

2) Assign new data to the resulting clusters;

I wanted to highlight the boundaries of clusters as min/max values ​​for each coordinate of an observation belonging to the cluster, then assign observations from the new data to a particular cluster in accordance with its boundaries. However, the problem is that cluster boundaries intersect and each observation can belong to several clusters. What adequate methods can be used to discriminate data in this case?