To define an "absolute" filling radius in a situation where "M" is equipped with a Riemannian metric "g", Gromov proceeds as follows. One exploits an imbedding due to C. Kuratowski. One imbeds "M" in the Banach space "L"∞("M") of bounded Borel functions on "M", equipped with the sup norm formula_9. Namely, we map a point "x" ∈ "M" to the function "fx" ∈ "L"∞("M") defined by the formula "fx(y)" = "d(x,y)" for all "y" ∈ "M", where "d" is the distance function defined by the metric. By the triangle inequality we have formula_10 and therefore the imbedding is strongly isometric, in the precise sense that internal distance and ambient distance coincide. Such a strongly isometric imbedding is impossible if the ambient space is a Hilbert space, even when "M" is the Riemannian circle (the distance between opposite points must be "π", not 2!). We then set "E" = "L"∞("M") in the formula above, and define