DeclareMathOperator*{argmin}{arg,min}

[bm theta_c^p coloneqq  adjustlimitsargmin_{bm theta_c in mathbb{R}^{n_{p} } }  max_{bm theta in mathcal{D}_{bm theta}} left| bm theta - bm theta_c right|_p ]

RequirePackage{fix-cm}
documentclass{article}
usepackage{mathtools,amssymb}
usepackage{bm}

DeclareMathOperator*{argmin}{arg,min}
DeclareMathSizes{5}{5}{4}{4}


begin{document}

[
bm{theta}_c^p coloneqq
adjustlimitsargmin_{bm{theta}_c in mathbb{R}^{n_{p}}}
             max_{bm{theta} in mathcal{D}_{bm theta}}
             lVertbm{theta} - bm theta_c rVert_p
]
[
bm{theta}_c^p coloneqq
adjustlimitsargmin_{bm{theta}_c in mathbb{R}^{text{$n_{p}$}}}
             max_{bm{theta} in mathcal{D}_{bm theta}}
             lVertbm{theta} - bm theta_c rVert_p
]

end{document}

max { (x-3) : x in [-1,1] }

documentclass{article}

usepackage{mathtools,amssymb,bm}

DeclareMathOperator*{argmin}{arg,min}
DeclarePairedDelimiter{norm}{lVert}{rVert}

newcommand{btheta}{bm theta}

begin{document}

begin{equation*}
  btheta_c^p coloneqq
  argmin
  Bigl{
    max_{btheta in mathcal{D}_{btheta}} norm{btheta - btheta_c}_p
    : btheta_c in mathbb{R}^{n_{p} }
  Bigr}
end{equation*}

end{document}