A solution using mdframed and allowing page breaks; simply changing the value for \myRad you can get different separations and values for the radius of the arc:
\documentclass{article}
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\newlength\myRad
\setlength\myRad{20pt}
\newmdenv[
settings={\tikzset{line cap=rect}},
innerrightmargin=2\myRad,
innerleftmargin=2\myRad,
innertopmargin=2\myRad,
innerbottommargin=2\myRad,
singleextra={
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=-\myRad,xshift=2\myRad]P-|O) -- ([yshift=-\myRad,xshift=-2\myRad]P);
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=-2\myRad,xshift=-\myRad]P) -- ([yshift=2\myRad,xshift=-\myRad]P|-O);
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=2\myRad,xshift=\myRad]O) -- ([yshift=-2\myRad,xshift=\myRad]O|-P);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=\myRad,yshift=-2\myRad]P-|O) arc[start angle=-90,end angle=0] (P-|O);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad,yshift=-2\myRad]P) arc[start angle=270,end angle=180] (P);
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=\myRad,xshift=2\myRad]O) -- ([yshift=\myRad,xshift=-2\myRad]P|-O);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=\myRad,yshift=2\myRad]O) arc[start angle=90,end angle=0] (O);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad,yshift=2\myRad]P|-O) arc[start angle=90,end angle=180] (P|-O);
},
middleextra={
\draw[gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad]P) -- ([xshift=-\myRad]P|-O);
\draw[gray!30,line width=5pt,line cap=rect] ([xshift=\myRad]O) -- ([xshift=\myRad]O|-P);
},
firstextra={
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=-\myRad,xshift=2\myRad]P-|O) -- ([yshift=-\myRad,xshift=-2\myRad]P);
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=-2\myRad,xshift=-\myRad]P) -- ([xshift=-\myRad]P|-O);
\draw[gray!30,line width=5pt,line cap=rect] ([xshift=\myRad]O) -- ([yshift=-2\myRad,xshift=\myRad]O|-P);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=\myRad,yshift=-2\myRad]P-|O) arc[start angle=-90,end angle=0] (P-|O);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad,yshift=-2\myRad]P) arc[start angle=270,end angle=180] (P);
},
secondextra={
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=\myRad,xshift=2\myRad]O) -- ([yshift=\myRad,xshift=-2\myRad]P|-O);
\draw[gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad]P) -- ([yshift=2\myRad,xshift=-\myRad]P|-O);
\draw[gray!30,line width=5pt,line cap=rect] ([yshift=2\myRad,xshift=\myRad]O) -- ([xshift=\myRad]O|-P);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=\myRad,yshift=2\myRad]O) arc[start angle=90,end angle=0] (O);
\draw[radius=\myRad,gray!30,line width=5pt,line cap=rect] ([xshift=-\myRad,yshift=2\myRad]P|-O) arc[start angle=90,end angle=180] (P|-O);
}
]{invcor}
\begin{document}
\begin{invcor}
\lipsum[2]
\end{invcor}
\begin{invcor}
\lipsum[1-7]
\end{invcor}
\end{document}

Update:
I introduced three mdf keys myRad, myLineWd, and myLineColor allowing you to further customize the attributes (of course, frames still admit page breaks):
\documentclass[dvipsnames]{article}
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\makeatletter
\newlength\mdf@@myRad
\newlength\mdf@@myLineWd
\define@key{mdf}{myRad}{\global\setlength\mdf@@myRad{#1}}
\define@key{mdf}{myLineWd}{\global\setlength\mdf@@myLineWd{#1}}
\define@key{mdf}{myLineColor}{\global\def\mdf@@myLineColor{#1}}
\newmdenv[
myRad=20pt,
myLineWd=5pt,
myLineColor=gray!30,
innerrightmargin=2\mdf@@myRad,
innerleftmargin=2\mdf@@myRad,
innertopmargin=2\mdf@@myRad,
innerbottommargin=2\mdf@@myRad,
singleextra={
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=-\mdf@@myRad,xshift=2\mdf@@myRad]P-|O) -- ([yshift=-\mdf@@myRad,xshift=-2\mdf@@myRad]P);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=-2\mdf@@myRad,xshift=-\mdf@@myRad]P) -- ([yshift=2\mdf@@myRad,xshift=-\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=2\mdf@@myRad,xshift=\mdf@@myRad]O) -- ([yshift=-2\mdf@@myRad,xshift=\mdf@@myRad]O|-P);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad,yshift=-2\mdf@@myRad]P-|O) arc[start angle=-90,end angle=0] (P-|O);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad,yshift=-2\mdf@@myRad]P) arc[start angle=270,end angle=180] (P);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=\mdf@@myRad,xshift=2\mdf@@myRad]O) -- ([yshift=\mdf@@myRad,xshift=-2\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad,yshift=2\mdf@@myRad]O) arc[start angle=90,end angle=0] (O);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad,yshift=2\mdf@@myRad]P|-O) arc[start angle=90,end angle=180] (P|-O);
},
middleextra={
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad]P) -- ([xshift=-\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad]O) -- ([xshift=\mdf@@myRad]O|-P);
},
firstextra={
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=-\mdf@@myRad,xshift=2\mdf@@myRad]P-|O) -- ([yshift=-\mdf@@myRad,xshift=-2\mdf@@myRad]P);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=-2\mdf@@myRad,xshift=-\mdf@@myRad]P) -- ([xshift=-\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad]O) -- ([yshift=-2\mdf@@myRad,xshift=\mdf@@myRad]O|-P);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad,yshift=-2\mdf@@myRad]P-|O) arc[start angle=-90,end angle=0] (P-|O);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad,yshift=-2\mdf@@myRad]P) arc[start angle=270,end angle=180] (P);
},
secondextra={
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=\mdf@@myRad,xshift=2\mdf@@myRad]O) -- ([yshift=\mdf@@myRad,xshift=-2\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad]P) -- ([yshift=2\mdf@@myRad,xshift=-\mdf@@myRad]P|-O);
\draw[\mdf@@myLineColor,line width=\mdf@@myLineWd,line cap=rect] ([yshift=2\mdf@@myRad,xshift=\mdf@@myRad]O) -- ([xshift=\mdf@@myRad]O|-P);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=\mdf@@myRad,yshift=2\mdf@@myRad]O) arc[start angle=90,end angle=0] (O);
\draw[\mdf@@myLineColor,radius=\mdf@@myRad,line width=\mdf@@myLineWd,line cap=rect] ([xshift=-\mdf@@myRad,yshift=2\mdf@@myRad]P|-O) arc[start angle=90,end angle=180] (P|-O);
},
]{invcor}
\makeatother
\begin{document}
\begin{invcor}
\lipsum[4]
\end{invcor}
\begin{invcor}[
myLineColor=Dandelion,
myLineWd=3pt]
\lipsum[4]
\end{invcor}
\begin{invcor}[
myRad=10pt,
innerleftmargin=20pt,
innerrightmargin=20pt,
innertopmargin=20pt,
innerbottommargin=20pt,
myLineColor=Maroon,
linecolor=MidnightBlue,
linewidth=2pt,
myLineWd=3pt]
\lipsum[4]
\end{invcor}
\end{document}
