# Characterizing surface area

(This question is a variant of an unanswered question at math.stackexchange.)

The Definition section of Wikipedia's article on surface area currently starts as follows:

While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function $$S \mapsto A(S)$$ which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface $S$ is a union of finitely many pieces $S_1, \dots, S_r$ which do not overlap except at their boundaries, then $$A(S) = A(S_1) + \cdots + A(S_r).$$ Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called ''piecewise smooth''.

It seems that not all the properties being hinted at are mentioned explicitly.

Indeed, I can define the "constant-curvature surface area" of $S$ to be

the supremum of the sums of surface areas of finitely many disjoint subsets of $S$, all of which have constant curvature,

and it will satisfy all the mentioned properties. But the constant-curvature surface area of a general ellipsoid, say, will be zero. So:

Which properties are missing in the Wiki article?

• We certainly want some sort of approximation continuity property, which is not satisfied by the Schwarz lantern. – Ben McKay Apr 11 '16 at 19:50
• What about adding the property of positivity: if S contains the (diffeomorphic) image of a 2-disc of positive radius then A(S)>0? It should ensure that the local area is not too far from the value of the well-known formula with double integrals... – Loïc Teyssier Apr 11 '16 at 20:13
• @LoïcTeyssier it is not enough, say you can take an integral of $|K|+1$, where $K$ denotes Gauss curvature. – Anton Petrunin Apr 11 '16 at 20:57
• See my answer here for Busemann's axioms on triangle area, especially the second: mathoverflow.net/questions/164053/… – Matt F. Apr 11 '16 at 21:55

Say if there is a distance nonexpanding map between surfaces $f\colon S\to S'$ then $$\mathop{\rm area} S\ge \mathop{\rm area} S'.$$