Chebyshev distance (L-infinity norm)

For a given pair of vectors and , the Chebyshev distance is the largest difference between two corresponding components:

As hinted in the notation, the Chebyshev distance is also the norm. Recall that the L-p norm is defined as

As becomes large, the sum is increasingly dominated by the dimension with the greatest absolute difference between and . In other words, if , then

Hence,

As the inner and outer exponents cancel, we get the intuition that an norm should match the definition given above. This can be proven more rigorously by rewriting the definition of the norm as

where again . Factoring out and canceling the exponents, we find that the remaining term simplifies to 1 in the limit where .