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](https://mathoverflow.net/q/352421/121665). 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.



Aleksandrov's proof of the second order differentiability of convex functions
The paper [On the second differentiability of convex surfaces](https://link.springer.com/article/10.1007/BF00150866) 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. 


