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

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.

• There's a paper by Bianchi, Colesanti, and Pucci titled "On the second differentiability of convex surfaces" which gives brief synopses of different approaches to the theorem of Alexandroff (including the preceding two dimensional case due to Busemann and Feller). Maybe instead of reading Russian/German the proof can be pieced together from the descriptions there? Mar 2, 2020 at 3:21
• @WillieWong If you turn it into an answer, I will accept it (unless someone writes a more detailed one). Mar 2, 2020 at 14:16
• Sorry to bother. Do you have an electronic version of the reference [A]? I can't find it online. Thank you very much! Feb 25 at 9:45
• @foxell Unfortunately, I don't know where to find this paper. Feb 25 at 14:29
• @PiotrHajlasz Never mind--there is no English version online. <mathoverflow.net/questions/353950/…> Feb 25 at 14:50

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.