How to make expectation symbol and max symbol line up

Question

Suppose you prefer the max symbol to be centered (height-wise) compared to a capitol letter that is taller. How would you do that. For example, below

\documentclass[14pt]{beamer}
\usetheme{Madrid}

\begin{document}
  \begin{frame}
    \begin{equation*}
         \max_{r_t} \left\{ 
         \mathbb{E} \left[ \sum_{t=0}^{\infty} r_t^t \right]
         \right\}
    \end{equation*}
  \end{frame}
\end{document}

I would like the max symbol moved slightly up so it lines up with the center of the expectation symbol.

Answer 1

Rather than raise the height of \max_{r_t}, I would lower the \mathbb{E} particle, making it look more like a purely symbolic entity (rather than an uppercase letter). One way to do this is to encase it in a \vcenter{\hbox{...}} set of directives.

Note that this method (as well as the one you initially proposed) "works", typographically speaking, only if there's a large symbol (such as \sum and \int) nearby. If the surrounding material is mostly text aligned on a common baseline, lowering the particle \mathbb{E} would probably look bizarre.

Incidentally, I would not use \left and \right to size the fence symbols in this formula, as doing so results in fences that are too large and thus dominate the material they enclose. Instead of \left and \right, I would use \biggl and \biggr, striking a balance between the material enclosed by the fences and the fences themselves.

\documentclass[14pt]{beamer}
\usetheme{Madrid}
\begin{document}
  \begin{frame}
    \[
         \max_{r_t} \biggl\{ 
         \mathbb{E}
         \biggl[ \, \sum_{t=0}^{\infty} r_t^t \biggr]
         \biggr\}
      \quad\text{vs.}\quad
         \max_{r_t} \biggl\{ 
         \vcenter{\hbox{$\mathbb{E}$}}
         \biggl[ \, \sum_{t=0}^{\infty} r_t^t \biggr]
         \biggr\}
    \]
  \end{frame}
\end{document}

Answer 2

You seem to want the expectation symbol is slightly moved down, but my advice is: don't. Letters are expected to hang on the baseline, just like max does.

The method is simply to say

\mathop{\mathbb{E}}\nolimits

because a one letter \mathop is automatically centered around the formula axis (like \sum is).

However, the same example shows why this is not really a good idea.

\documentclass{article}

\usepackage{amsmath,amssymb}

\newcommand{\expect}{\mathop{\mathbb{E}}\nolimits}

\begin{document}
\[
\max_{r_t} \biggl\{ 
  \mathbb{E}
  \biggl[ \, \sum_{t=0}^{\infty} r_t^t \biggr]
\biggr\}
\quad\text{vs.}\quad
\max_{r_t} \biggl\{ 
  \expect
  \biggl[ \, \sum_{t=0}^{\infty} r_t^t \biggr]
\biggr\}
\quad\text{vs.}\quad
\expect(a)
\]
\end{document}