I this paper which is titled "KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation", in the section about "kmeans algorithm for vector quantization", there is the following paragraph:
we denote the codebook matrix by $C=[c_1,c_2,\dots,c_K]$ the codewords being the columns. when C is given, each signal is represented as its closest codeword (under $l_2$-norm distance) we can write $y_i=Cx_i$ where $x_i=e_j$ is a vector from the trivial basis, with all zero entries except a one in the jth position. the index j is selected such that $$\forall_{k \neq j} ||y_i-Ce_j||_2^2 \leq||y_i-Ce_k||_2^2 \\$$
I know how kmeans algorithm works and what does it do, the thing I don't understand is the above inequality and the selection of j.
The vector $Ce_k$ is the $k$th column of the matrix $C$ for $k = 1,\ldots,K$. So this inequality says that $Ce_j$, that is, the $j$th column of $C$, is closer to $y_i$ in the $l_2$-norm then all other columns of $C$. Another way of writing this is that $j$ is the index such that
$$ \|y_i - c_j\|_2^2 = \min_k \|y_i - c_k\|_2^2, $$
where $c_k$ is the $k$th column of $C$.