There is a sentence in Oded Regev'lecture note that "$0$ is part of any lattice and hence the closest vector to $0$ is $0$ itself!". I'm having trouble understanding it. Can someone help me understand it?
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1 Answer
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Taking each basis vector with coefficient 0 gives you the zero point $(0,\ldots,0)$. If you ask what is closest point to it in the lattice, then the answer is the zero point itself, with distance 0.
You can of course ask what is the second closest point to 0 in the lattice, and it is not difficult to see that it corresponds to the shortest (non-zero) vector in the lattice.
In fact, this discussion applies to any point in the lattice itself.
