horizontal margin in tabu

Question

I have this code ;

\documentclass[oneside]{oblivoir}
\usepackage{amsmath,amssymb,amsthm,mathrsfs,amsthm}
\usepackage{paralist,kotex,caption,mdframed,tabu,graphicx}

\begin{document}

\(g\) is a holomorphic function on the compact set \(\triangle\).
By the maximum modulus theorem, \(|g|\) attains its maximum on the boundary of \(\triangle\).
By symmetry of \(\triangle\), \(|g|\) attains its maximum on the segment \(ab\), where the segment can be parametrized by \(\gamma(t):[-1,1]\to\mathbb C\) as follows;
\[\gamma(t)=t\quad(-1\le t\le1)\]
The value of \(g\) on the segment can be written as
\[g(\gamma(t))=(t^2-1)(t-\sqrt3i)=t(t^2-1)-\sqrt3(t^2-1)i\]
and
\[\left|g(\gamma(t))\right|^2=t^2(t^2-1)^2+3(t^2-1)^2=(t^2-1)^2(t^2+3).\]
Since
\[
\frac{d}{dt}\left|g(\gamma(t))\right|^2
=2(t^2-1)\cdot2t(t^2+3)+(t^2-1)^2\cdot2t=2t(t^2-1)(3t^2+5)
\]
\(\left|g(\gamma(t))\right|^2\) has critical values(local maximum and local minimum) at \(t=-1\), \(t=0\), \(t=1\) where
\begin{center}
\tabulinesep=5pt
\begin{tabu}to.9\textwidth{X[1.5$c]|X[$c]|X[$c]|X[$c]|X[$c]|X[$c]}\toprule
t                           &-1                     &\cdots     &0          &\cdots     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &\vbox{local minimum}   &\nearrow   &\vbox{ local maximum}  &\searrow   &\vbox{ local minimum}\\\bottomrule
\end{tabu}
\end{center}
Therefore, \(|g|\) has maximum value \(\sqrt3\) on the whole triangle \(\triangle\).
\end{document}

with the ugly result looks like

I don't want to enlarge the width of the second column, but want the word 'local minimum' be written in two lines, not three lines.

I tried to change the size of the font(normalsize, small, footnotesize, scriptsize, tiny), unsuccessfully even when the size is 'tiny'.

Is there any parameter like 'tabulinesep' or 'extrarowsep' to set the horizontal(not vertical) margin inside the cell?

Answer 1

As already mentioned in the comments, I'd recomment to not use tabu. Instead, here are a bunch of different versions of your table. In these I have kept the combination of horizontal and vertical lines. With the help of cellspace I have added some vertical space around the horizontal lines.

In the first three examples, I have used a simple tabular environment and manually determined the required width for the cells. Since you seem to prefer a more automated solution, I have added another set of three examples all making use of tabularx which also allows to specify an overall width of the resulting table.

\documentclass[oneside]{oblivoir}

\usepackage{cellspace}
\setlength\cellspacetoplimit{6pt}
\setlength\cellspacebottomlimit{6pt}


\begin{document}

\begin{center}
\setlength{\tabcolsep}{5.5pt}
\begin{tabular}{|>{$}Sc<{$}|Sc*{2}{|>{$}Sc<{$}|Sc}|}
\hline
t                           &-1                     &\cdots     &0          &\cdots     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &local minimum   &\nearrow   & local maximum  &\searrow   & local minimum\\\hline
\end{tabular}
\end{center}


\begin{center}
\setlength{\tabcolsep}{5.5pt}
\begin{tabular}{|>{$}Sc<{$}*{5}{|>{\raggedright\arraybackslash}S{p{1.53cm}}}|}
\hline
t                           &-1                     & $\cdots$     &0          & $\cdots$     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &local minimum   & $\nearrow$   & local maximum  & $\searrow$   & local minimum\\\hline
\end{tabular}
\end{center}


\begin{center}
\setlength{\tabcolsep}{5.5pt}
\begin{tabular}{|>{$}Sc<{$}*{5}{|>{\centering\arraybackslash}S{m{1.53cm}}}|}
\hline
t                           &-1                     & $\cdots$     &0          & $\cdots$     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &local minimum   & $\nearrow$   & local maximum  & $\searrow$   & local minimum\\\hline
\end{tabular}
\end{center}


\begin{center}
\setlength{\tabcolsep}{0pt}
\begin{tabularx}{0.8\textwidth}{|>{$}Sc<{$}*{5}{|>{\centering\arraybackslash}X}|}
\hline
t                           &-1                     & $\cdots$     &0          & $\cdots$     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &local minimum   & $\nearrow$   & local maximum  & $\searrow$   & local minimum\\\hline

\end{tabularx}
\end{center}

\begin{center}
\setlength{\tabcolsep}{3pt}
\begin{tabularx}{0.8\textwidth}{|>{$}Sc<{$}*{5}{|>{\centering\arraybackslash}X}|}
\hline
t                           &-1                     & $\cdots$     &0          & $\cdots$     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                & \footnotesize local minimum   & $\nearrow$   & \footnotesize local maximum  & $\searrow$   & \footnotesize local minimum\\\hline

\end{tabularx}
\end{center}

\begin{center}
\setlength{\tabcolsep}{5.5pt}
\begin{tabularx}{\textwidth}{|>{$}Sc<{$}*{5}{|>{\centering\arraybackslash}X}|}
\hline
t                           &-1                     & $\cdots$     &0          & $\cdots$     &1  \\\hline
\frac d{dt}|g(\gamma(t))|^2 &0                      &+          &0          &-          &0  \\\hline
|g(\gamma(t))|^2                &local minimum   & $\nearrow$   & local maximum  & $\searrow$   & local minimum\\\hline

\end{tabularx}
\end{center}
\end{document}

Answer 2

In case, that you still persist to use tabu, than the following solution may help you:

\documentclass{article} % I haven't your document class
\usepackage{tabu} % used is version v2.9

\begin{document}
\begin{center}
\tabulinesep=5pt
\setlength\tabcolsep{4pt}
\begin{tabu}to.8\textwidth{X[1.5$c]|X[1.5$cm]|X[$c]|X[$c]|X[$c]|X[$c]}
    \tabucline [1pt]{-}
t                           &-1             &\cdots     & 0 & \cdots    &1  \\
    \hline
\frac d{dt}|g(\gamma(t))|^2 &0              &+          & 0 & -         &0  \\
    \hline
|g(\gamma(t))|^2            & \vbox{local 
                                    minimum}& \nearrow  &   & \searrow  &\\
    \tabucline [1.5pt]{-}
\end{tabu}
\end{center}
\end{document}

Addendum: With tabularx instead of tabu, using makecell with fix for use in m column types and mathtools for defining delimiters, considering edited question, the above solution can be rewritten into:

\documentclass{article}
\usepackage{mathtools}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\usepackage{makecell}
\makeatletter %% fix of incompability between \makegapedcells and m column type
              %% provided by Ulrike Fisher: http://tex.stackexchange.com/questions/319768/
\renewcommand\mcell@classz{\@classx
   \@tempcnta \count@
   \prepnext@tok
   \@addtopreamble{%\mcell@mstyle
      \ifcase\@chnum
         \hfil
         \mcell@agape{\d@llarbegin\insert@column\d@llarend}\hfil \or
         \hskip1sp
         \mcell@agape{\d@llarbegin\insert@column\d@llarend}\hfil \or
         \hfil\hskip1sp
         \mcell@agape{\d@llarbegin \insert@column\d@llarend}\or
         \mcell@agape{$\vcenter
         \@startpbox{\@nextchar}\insert@column\@endpbox$}\or
         \mcell@agape{\vtop
         \@startpbox{\@nextchar}\insert@column\@endpbox}\or
         \mcell@agape{\vbox
         \@startpbox{\@nextchar}\insert@column\@endpbox}%
      \fi
      \global\let\mcell@left\relax\global\let\mcell@right\relax
    }\prepnext@tok}
    \makeatletter
\usepackage{tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}

\begin{document}
\begin{center}
\setcellgapes{5pt}
\makegapedcells
\setlength\tabcolsep{4pt}
\renewcommand\tabularxcolumn[1]{m{#1}}  % <---
\begin{tabularx}{.9\textwidth}{@{} >{$}c<{$} | C | >{$}C<{$} | C | >{$}C<{$} | C @{}} % <---
    \Xhline{1pt}
t       & $-1$          & \cdots    & 0             & \cdots    & 1             \\
    \hline
\frac{d}{dt}\abs{g(\gamma(t))}^2 
        & 0             & +         & 0             & -         & 0             \\
    \hline
\abs*{g(\gamma(t))}^2   
        & local minimum & \nearrow  & local minimum & \searrow  & local minimum \\
    \Xhline{1pt}
\end{tabularx}
\end{center}
\end{document}

which gives: