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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite [[graph|0JD5C82QMB]], let $\\delta \\colon E \\to [0, \\infty)$ be a weight on its edges, and let $k \\ge 1$ be an integer.\nA *realization* of $(G, \\delta)$ in $\\R^k$ is a map $p \\colon V \\to \\R^k$ such that\n$$\\norm{p(u) - p(v)} = \\delta(uv) \\quad \\text{for every edge } uv \\in E.$$\nHere $\\norm{p(u) - p(v)}$ is the [[Euclidean distance|0C95TG69SG]] of the two points.\nThe *distance geometry problem* asks, for given $(G, \\delta)$ and $k$, whether a realization in $\\R^k$ exists, and to find one.\nDistance geometry is the study of such realizations and of the point sets they describe.\n\n## Remarks\n\nExample: let $G$ be a triangle with edge weights $3$, $4$ and $5$, and $k = 2$.\nThe points $(0, 0)$, $(3, 0)$ and $(3, 4)$ form a realization, since their pairwise distances are $3$, $4$ and $5$.\n\nNon-example: the same triangle with weights $1$, $1$ and $3$ has no realization in any $\\R^k$, since $3 \u003e 1 + 1$.\n\nWhen $G$ is a complete graph, a realization exists in some $\\R^k$ if and only if the squared weights form a [[Euclidean distance matrix|1BCJ7A73KS]].\nA realization is never unique: an isometry of $\\R^k$ moves it to another one.\n",
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