Prescription of Gauss curvature using optimal mass transport

Abstract

In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).

Appendix

In this appendix, we prove the following result

Lemma 4.10

Let $\mu$ be a finite Borel measure on the unit sphere ${\mathbb {S}}^m$ endowed with its canonical distance d. Suppose that $\mu$ is absolutely continuous with respect to the standard uniform measure on ${\mathbb {S}}^m$ and $\mu ({\mathbb {S}}^m)$ is a positive rational number. Then for any $\alpha >0$, there exists a finite partition $(P_i)_{1\le i\le K}$ of ${{\mathbb {S}}}^m$ (depending on $\alpha$) such that for all i, $\mu (P_i)>0$ is a rational number and $diam (P_i) < \alpha$.

Remark 4.11

When $\mu$ is the uniform probability measure $\sigma$, the proof below together with the expression of $\sigma$ in polar coordinates guarantee that we can further require $\forall i, \;\sigma (P_i)= 1/M$, being M a sufficiently large integer.

Proof

The proof is by induction on the dimension m. For $m=1$, fix a number $\alpha _1>0$. Then, partition ${\mathbb {S}}^1$ into finitely many left-open, right-closed segments $(I_j)_{1\le j\le K_1}$ whose length $l(I_j)$ satisfies $l(I_j)<\alpha _1$ and $\mu (I_j) \in {{\mathbb {Q}}}$ (we use that $s \mapsto \mu ((a,s])$ is continuous); $\mu (I_{K_1}) \in {{\mathbb {Q}}}$ is guaranteed by $\mu ({\mathbb {S}}^1)\in {{\mathbb {Q}}}$. For $m=2$, fix a point $N\in {\mathbb {S}}^2$ and $\alpha _2>0$. Consider a partition $(C_i)_{1 \le i \le K_2}$ where $C_1$ is the closed ball with radius $R_1$ and center N, $C_i= \{z\in {\mathbb {S}}^2; R_{i-1} < d(N,z) \le R_i\}$ for $i \in \{2,\ldots ,K_2-1\}$ and $C_{K_2}$ is the closed ball with radius $\pi -R_{K_2}$ and center $-N$. We require that the $(R_i)$ satisfy:

$$\begin{aligned} \alpha _2/2 < R_1< \alpha _2, \quad \alpha _2/2 < R_i-R_{i-1} < \alpha _2, \quad \pi -R_{K_2} < \alpha _2 \end{aligned}$$

and $\mu (C_i)\in {{\mathbb {Q}}}$. Let us set p the projection onto the equator relative to N. Note that the measures are absolutely continuous with respect to the uniform measure on the circle thus, applying the case $m=1$ to all the measures , we get a partition $(P_i)_{1\le i\le K}$ of ${\mathbb {S}}^2$ (namely $(C_i \cap p^{-1}(I_j^i))_{i,j}$, being $(I_j^i)_j$ the partition corresponding to ) such that $\mu (P_i) \in {{\mathbb {Q}}}$. Moreover, the expression of the spherical distance in polar coordinates implies that the diameter of any $P_i$ is smaller than $\alpha$ provided $\alpha _1$ and $\alpha _2$ are sufficiently small. The higher dimensional case easily follows from the arguments used for $m=2$. $\square$