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Is there the name for an algorithm of cluster assignment that is based uniquely on the distance between the data point to classify and the center of the cluster?

Let me be more clear:

Let's say that I have two clusters $A,B$ made by $N$ points $x(i)_A$ and $x(j)_B$. I have to assign a new point $x_{new}$ to it's cluster.

The point is assigned to the cluster that minimize the distance between the point and the center of the cluster:

$$ min_{A,B} ( d_A(x_{new}, \frac{1}{N}\sum_{i \in A}x_i), d_B(x_{new}, \frac{1}{N}\sum_{j \in B}x_j) ) $$

Does this procedure has a name? Is there an algorithm similar to this? The closest algorithm that I can think of is k-Nearest Neighbour, but it's easy to think of cases where kNN performs properly and this algorithm performs very poorly. For instance, this algorithm as such will perform badly on data shaped non-uniformly / non-gaussian.

This question arise from a paper that offer a procedure for doing supervised quantum machine learning, and claims that:

Consider the task of assigning a post-processed vector $u \in R^n$ to one of two sets $V, W$, given $M$ representative samples $v_j ∈ V$ and $M$ samples $w_k ∈ W$. A common method for such an assignment is to evaluate the distance $|u−\frac{1}{M}\sum_{j} v_j |$ between $u$ and the mean of the vectors in $V$ , and to assign $u$ to $V$ if this distance is smaller than the distance between $u$ and the mean of $W$.

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K means clustering kind of works like this.
It uses the Lloyd algorithm, which basically has two steps.
Per each new sample:
1) Choose the best cluster, by calculating the minimum distance to available clusters mean.
2) Recalculate chosen cluster mean.

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