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April 27

Generalization of perimeter and surface area to higher dimensions

Is there a known name for the generalization of perimeter and surface area to higher dimensions, which produces an (n − 1)-dimensional quantity from an n-dimensional figure? GeoffreyT2000 (talk) 14:43, 27 April 2016 (UTC)[reply]

I would call it the "edge", as in the "edge length" (perimeter) of a circle or the "edge area" (surface area) of a sphere or "edge volume" of a 4D object. However, edge (geometry) defines that term far more restrictively. Boundary (topology) may be the proper term. (For proof that my use of the word "edge" isn't totally off the wall (although it is "outside the box"), see Manifold#Manifold_with_boundary, where they seem to use it as a synonym for "boundary".) StuRat (talk) 14:51, 27 April 2016 (UTC)[reply]

The standard term is "boundary". I've never heard "edge" used in this way. There are various ways of measuring it to get a number. One is the Hausdorff measure, which can be defined for quite general sets. If the boundary is rectifiable, then one can calculate the measure as an integral, in a manner analogous to the calculation of arc length and surface area in basic multivariable calculus. Sławomir
Biały 15:04, 27 April 2016 (UTC)[reply]

From Hypersurface: "Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface." So would the correct term be the hypervolume of the object's hypersurface? Loraof (talk) 16:05, 27 April 2016 (UTC)[reply]

Hypervolume, though the term is rarely used, generally means the n-dimensional volume. I think the standard term is

(n-1)

-dimensional measure. Often this implicitly means the Hausdorff measure, but it can sometimes mean other things like Jordan content. Sławomir
Biały 19:20, 27 April 2016 (UTC)[reply]

Some "volumes" in various dimensions are described at Mixed volume. Staecker (talk) 14:12, 28 April 2016 (UTC)[reply]

In geometric measure theory, people typically just use the word "perimeter" for the (n-1)-dimensional Hausdorff measure of the boundary of a set. Compare Caccioppoli set. —Kusma (t·c) 18:30, 28 April 2016 (UTC)[reply]