Aleksandrov's proof of the second order differentiability of convex functions

Question

Aleksandrov [A], proved a remarkable property of convex functions.

Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and a symmetric $(n\times n)$ matrix $D^2f(x)$ such that $$ \lim_{y\to x} \frac{|f(y)-f(x)-Df(x)(y-x)-\frac{1}{2}(y-x)^TD^2f(x)(y-x)|}{|y-x|^2}=0. $$

I know two proofs of this result. One based on the theory of maximal monotone functions and one based on the fact that the second order distributional derivatives of a convex function are Radon measure. Both proofs are mentioned in Second order differentiability of convex functions. Since these proofs use relatively modern techniques not available during Aleksandrov's time, his argument must have been very different.

Question 1. Can you briefly explain what was the idea of the original proof due to Aleksandrov?

My guess would be that his proof was based on methods of differnetial geometry. What else could he use in those days?

Question 2. In there any textbook where I can find the original proof due to Aleksandrov?

[A] A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. (Russian) Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6, (1939), 3–35.

Answer 1

The paper On the second differentiability of convex surfaces by Bianchi, Colesanti, and Pucci (Geometriae Dedicata volume 60, pages 39–48 (1996)) concerns the proof of the Busemann-Feller-Alexandroff Theorem on the second order differentiability of convex functions. Its introduction gives brief synopses on several different methods of proof, including the original argument of Busemann-Feller and later Alexandroff, the two methods you mentioned in the other question (the monotone operator method of Rockafeller (using a result of Mignot); and the measure/distribution method of Reshetnyak), as well as a different one by Bangert (using almost purely differential geometric methods).

The paper gives also a new proof of the theorem, which is claimed to be in the same spirit of the original arguments of Busemann-Feller and Alexandroff. The authors considered the second order difference quotient of the convex function based at a point $x$, which they show has a limit a.e. as a convex function. This new convex function is related to the argument of Busemann-Feller in that the indicatrices constructed by Busemann-Feller are the 1-level-sets of this limited convex function.