Editing Distance geometry (section)

==== Semimetric space ====
Given a list of points on <math>R = \{P_0, \ldots, P_n\}</math>, <math>n \ge 0</math>, we can arbitrarily specify the distances between pairs of points by a list of  <math>d_{ij}> 0</math>, <math>0 \le i < j \le n</math>. This defines a [[Semi metric space|semimetric space]]: a metric space without [[triangle inequality]].

Explicitly, we define a semimetric space as a nonempty set <math>R</math> equipped with a semimetric <math>d: R\times R \to [0, \infty)</math> such that, for all <math>x, y\in R</math>,

# Positivity: <math>d(x, y) = 0</math> &nbsp; if and only if &nbsp;<math>x = y</math>.
# Symmetry: <math>d(x, y) = d(y, x)</math>.

Any metric space is [[Argumentum a fortiori|''a fortiori'']] a semimetric space. In particular, <math>\mathbb{R}^k</math>, the <math>k</math>-dimensional [[Euclidean space]], is the [[Canonical form|canonical]] metric space in distance geometry.

The triangle inequality is omitted in the definition, because we do not want to enforce more constraints on the distances <math>d_{ij}</math> than the mere requirement that they be positive.

In practice, semimetric spaces naturally arise from inaccurate measurements. For example, given three points <math>A, B, C</math> on a line, with <math>d_{AB} = 1, d_{BC} = 1, d_{AC} = 2</math>, an inaccurate measurement could give <math>d_{AB} = 0.99, d_{BC} = 0.98, d_{AC} = 2.00</math>, violating the triangle inequality.