On the history of the affine surface area for convex bodies
For a convex body \(K\) in \(n\)-space, whose boundary \(\partial K\) has everywhere positive Gauss curvature \(H_{n-1}\), W. Blaschke showed in 1923, that the integral over \(\partial K\) of \(H_{n-1}^{1/(n+1)}\), with respect to the Euclidean surface area measure on \(\partial K\), is invariant under the group of unimodular affine transformations. Blaschke called this the affine surface area of \(\partial K\), here denoted by \(O_{\text{aff}}(\partial K)\). This paper describes a number of subsequent definitions and representations of this invariant, as well as the extent to which they are more inclusive than those of Blaschke. Here is an example due to \textit{C. Schütt} and \textit{E. Werner} [Math. Scand. 66, No. 2, 275-290 (1990; Zbl 0739.52008)]. Let \(K_ \delta\) be the convex body, which is the common part of the intersections of \(K\) with those half-spaces, which meet \(K\) in a set of volume \(V(K)-\delta\) for \(\delta>0\) and small enough. Then \(O_{\text{aff}}(\partial K)\) is \(\lim_{\delta\to 0+} (V(K)-V(K_ \delta))/\delta^{2/(n+1)}\) up to a constant depending on \(n\) alone. Here the integration of \(H_{n- 1}^{1/(n+1)}\) is in Lebesgue's sense; this is justified by the existence almost everywhere on \(\partial K\) of the needed derivatives, as shown by A. D. Aleksandrov in 1939. The result holds for all \(K\) with interior points.
scientific article; zbMATH DE number 57246