What is Chebyshev distance (L∞)?

Chebyshev distance, also called L-infinity distance, measures the distance between two vectors as the single largest difference found in any one dimension, ignoring every other dimension’s difference entirely.

How is Chebyshev distance actually calculated between two vectors?

For each dimension, take the absolute value of the difference between the two vectors’ values in that dimension, just as Manhattan distance does, but instead of summing all of these differences together, Chebyshev distance simply keeps whichever single dimension produced the largest difference and discards the rest. The result is a distance driven entirely by the one dimension where the two vectors disagree the most, regardless of how closely they might agree across every other dimension.

Why is it called an “L-infinity” distance?

Euclidean distance, Manhattan distance, and Chebyshev distance are all part of a broader family of distance measures, generally distinguished by a parameter that controls how strongly larger differences get emphasized relative to smaller ones — Manhattan distance sits at one end of this family, treating every dimension’s difference equally regardless of size, and Euclidean distance sits further along it, squaring differences so that larger ones count disproportionately more. Chebyshev distance sits at the extreme far end of this same family: pushed to its limit, only the single largest difference ends up mattering at all, with every other dimension’s contribution effectively vanishing by comparison, which is exactly the behavior the “infinity” in its name refers to.

When does Chebyshev distance actually get used in practice?

Chebyshev distance is the natural choice whenever the practical question at hand is really about the single worst mismatch between two things, rather than their overall combined difference — deciding whether two configurations are “close enough” when every individual setting needs to fall within some tolerance, for instance, where a single wildly off setting should disqualify a match even if every other setting happens to line up perfectly. It’s a less common choice for typical vector search over embeddings, where the relationships across many dimensions collectively, rather than any single extreme dimension, usually carry the meaningful signal about similarity.

Chebyshev distance rounds out the L-p family of distance measures alongside Manhattan and Euclidean distance, both covered elsewhere in this glossary. From here, this site’s page on measuring distance and similarity between vectors places all of these choices in context to explain which one actually fits a given kind of data and problem.