In size theory, the natural pseudo-distance between two size pairs ${\displaystyle (M,\varphi :M\to \mathbb {R} )}$, ${\displaystyle (N,\psi :N\to \mathbb {R} )}$ is the value ${\displaystyle \inf _{h}|\varphi -\psi \circ h|_{\infty }}$, where ${\displaystyle h}$ varies in the set of all diffeomorphisms from the manifold ${\displaystyle M}$ to the manifold ${\displaystyle N}$ and ${\displaystyle \|\cdot \|_{\infty }}$ is the supremum norm. ${\displaystyle M}$, ${\displaystyle N}$ are assumed to be ${\displaystyle C^{1}}$ closed manifolds and the measuring functions ${\displaystyle \varphi ,\psi }$ are assumed to be ${\displaystyle C^{1}}$. Put another way, the natural pseudo-distance measures the infimum of the change of the measuring function induced by a diffeomorphism ${\displaystyle h:M\to N}$.

It can be proved ^[1] For a fixed length n, the Hamming distance is a metric on the vector space of the words of that length, as it obviously fulfills the conditions of non-negativity, identity of indiscernibles and symmetry, and it can be shown easily by complete induction that it satisfies the triangle inequality as well. The Hamming distance between two words a and b can also be seen as the Hamming weight of a−b for an appropriate choice of the − operator.

For binary strings a and b the Hamming distance is equal to the number of ones in a xor b. The metric space of length-n binary strings, with the Hamming distance, is known as the Hamming cube; it is equivalent as a metric space to the set of distances between vertices in a hypercube graph. One can also view a binary string of length n as a vector in ${\displaystyle \mathbb {R} ^{n}}$ by treating each symbol in the string as a real coordinate; with this embedding, the strings form the vertices of an n-dimensional hypercube, and the Hamming distance of the strings is equivalent to the Manhattan distance between the vertices.

^[2]

Pietro Donatini, Patrizio Frosini. Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.

Pietro Donatini, Patrizio Frosini. Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.

• Hamming weight
• Jaccard index
• Levenshtein distance (aka “edit distance”), a generalization of the Hamming distance
• Manhattan distance
• Similarity (mathematics)
• Similarity space on Numerical taxonomy
• Sørensen similarity index