An example in surface area.
For length and area, a central fact is that the value of the length of a curve or the area of a surface, as given by the Lebesgue theory, is at least as great as that given by the classical formula, whenever the latter has meaning. This is now found not to be valid in higher dimensions. We give an example of a continuous mapping of the unit cube into itself for which the value given by the formula exceeds the three-dimensional Lebesgue area of the corresponding suface.
Goffman
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DIVISION OF MATHEMATICAL SCIENCES, PURDUE UNIVERSITY.