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\begin{equation}
\begin{aligned}
\operatorname{res} &= -1 \cdot \sum_{i=1}^n \frac{1}{\lvert q_i \rvert} \cdot \lvert \frac{point.p - point_i.p}{\lvert p_i \rvert}\rvert 
\end{aligned}
\label{eq:closegap1}
\end{equation}

\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}

\abs{\frac{1}{2}}

\documentclass{book}
\usepackage{amsmath}

\begin{document}
\begin{equation} 
\begin{aligned} 
\operatorname{res} &= -1 \cdot \sum_{i=1}^n \frac{1}{\lvert q_i \rvert} \cdot \left\lvert% HIER!
 \frac{point.p - point_i.p}{\lvert p_i \rvert}\right\rvert%HIER!
 \\
\end{aligned} 
\end{equation}
\end{document}

\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{mathtools}

\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\begin{document}
  \( \abs{x} \)
  \[
    \abs*{\frac{a}{b}}
  \]

  \( \norm{x} \)
  \[
    \norm*{\frac{a}{b}}
  \]

\end{document}