Yesterday, while I am attending ICERM conference on fluid dynamics and turbluence, I learned from Nestor that there is another way to look at the Aleksandrov maximum principle via convex geometry. I appreciate his passion for sharing this knowledge with me.

In my previous blog post, Aleksandrov estimate plays an important role in regularity theory for equations in nondivergence form or fully nonlinear equations. For simplicity, we consider a convex function $h:D\rightarrow\mathbb{R}$ defined on a convex body $D$ satisfying $h=0$ on $\partial D$. Furthermore, we assume that the center of mass of $D$ is the origin. A classical Aleksandrov estimate is that

\begin{equation} \Vert h \Vert_{L^\infty(D)} \leq C_d|D|^{1/d}|\nabla h(D)| \end{equation} for some constant $C_d>0$.
There is an interesting connection with Blashcke-Santalo inequality and this estimate. To state this, let us define the polar dual \begin{equation*} D^*=\{x\in\mathbb{R}^d : x\cdot y \leq 1\quad \text{for all } y \in D\}. \end{equation*} In 1939, Mahler proved that \begin{equation*} 4^n (n!)^{-2}\leq |D| |D^*|\leq 4^n. \end{equation*} Later, Santalo proved that the upper bound has a precise geometric bound \begin{equation*} |D| |D^*|\leq |B_d|^2, \end{equation*} where $B_d$ is the $d$-dimensional unit ball. The inequality becomes equality when $D=B_d$. After this work, several interesting improvements were made to determine the precise lower bound for $|D| |D^*|$. I do not have a clear picture of how researchers could obtain optimal estimates by using the geometry of Banach spaces (this is slightly unclear to me). Bourgan-Milman proved the following estimate, which is now called Blashcke-Santalo inequality.

ongoing….