K-Means algorithm is a popular clustering method. However, it has two limitations: 1) it gets stuck easily in spurious local minima, and 2) the number of clusters

$k$

has to be given a priori. To solve these two issues, a multi-prototypes convex merging based K-Means clustering algorithm (MCKM) is presented. First, based on the structure of the spurious local minima of the K-Means problem, a multi-prototypes sampling (MPS) is designed to select the appropriate number of multi-prototypes for data with arbitrary shapes. Then, a merging technique, called convex merging (CM), merges the multi-prototypes to get a better local minima without

$k$

being given a priori. Specifically, CM can obtain the optimal merging and estimate the correct

$k$

. By integrating these two techniques with K-Means algorithm, the proposed MCKM is an efficient and explainable clustering algorithm for escaping the undesirable local minima of K-Means problem without given

$k$

first. Two theoretical proofs are given to guarantee that the cost of MCKM (MPS+CM) can achieve a constant factor approximation to the optimal cost of the K-Means problem. Experimental results performed on synthetic and real-world data sets have verified the effectiveness of the proposed algorithm.

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