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MOTIVATION Mines and factories Suppose that we have a collection of $n$ mines mining iron ore, and a collection of $n$ factories which consume the iron ore that the mines produce. Suppose for the sake of argument that these mines and factories form two Disjoint Subset s $M$ and $F$ of the Euclidean Plane $\backslash mathbb\{R\}^\{2\}$. Suppose also that we have a ''cost function'' $c\; :\; \backslash mathbb\{R\}^\{2\}\; imes\; \backslash mathbb\{R\}^\{2\}\; o+\; \backslash infty$, so that $c(x,\; y)$ is the cost of transporting one shipment of iron from $x$ to $y$. For simplicity, we ignore the time taken to do the transporting. We also assume that each mine can supply only one factory (no splitting of shipments) and that each factory requires precisely one shipment to be in operation (factories cannot work at half- or double-capacity). Having made the above assumptions, a ''transport plan'' is a Bijection $T\; :\; M\; o\; F$ --- i.e. an arrangement whereby each mine $m\; \backslash in\; M$ supplies precisely one factory $T(m)\; \backslash in\; F$. We wish to find the ''optimal transport plan'', the plan $T$ whose ''total cost'' :$c(T)\; :=\; \backslash sum\_\{m\; \backslash in\; M\}\; c(m,\; T(m))$ is the least of all possible transport plans from $M$ to $F$. Moving books: the importance of the cost function The following simple example illustrates the importance of the cost function in determining the optimal transport plan. Suppose that we have $n$ books of equal width on a shelf (the Real Line ), arranged in a single contiguous block. We wish to rearrange them into another contiguous block, but shifted one book-width to the right. Two obvious candidates for the optimal transport plan present themselves: # move all $n$ books one book-width to the right; ("many small moves") # move the left-most book $n$ book-widths to the right and leave all other books fixed. ("one big move")

u}^{-1} (s) ight) \, \mathrm{d} s.


Separable Hilbert spaces


Let $X$ be a Gaussian Measure on $X$ and $g\; (N)\; =\; 0$, then $\backslash mu\; (N)\; =\; 0$ also.

Let $\backslash mu\; \backslash in\; \backslash mathcal\{P\}\_\{p\}^\{r\}\; (x)$, $$
  :<math>r(x) x - x