The Isoperimetric Inequality and the Lebesgue Definition of Surface Area

The literature of these classical isoperimetric inequalities is very extensive (comprehensive presentations may be found in Blaschke [l] and Bonnesen [l](1)). In most instances, only convex curves and convex surfaces are considered, or else it is assumed that the curves and surfaces involved are sufficiently regular to permit the use of the classical formulas for the quantities a(C), 1(C), A(S), V(S). Briefly, the greater part of the literature relates to what may be termed the elementary range. Within the elementary range, the concepts involved in the inequalities (1) and (2) have generally accepted meanings, and the validity of these inequalities is a foregone conclusion, even though the actual proofs are of great interest and of substantial difficulty. On the other hand, the situation is quite different beyond the elementary range, especially in the case of the spatial isoperimetric inequality (2). It is well known that the number of formal definitions that have been proposed for surface area is very large. It is perhaps less well known that most of the more relevant definitions of surface area were found to conflict with each other in relatively simple non-elementary cases (see, for example, Nöbeling [l J). Similarly the concept of enclosed volume, involved in the inequality (2), admits of several plausible formal definitions which are readily seen to conflict with each other beyond the elementary range (cf. 1.4, 5.6, 5.8). Finally, the concept of closed surface lends itself to several fundamentally different interpretations (cf. Youngs [l, 2]). Consequently, beyond the elementary range the spatial isoperimetric inequality (2) is by no means an a priori obvious geometrical fact. Rather, this inequality may be construed as a test of adequate