%Blackboard Bold


\def\AA{{\mathbb A}}
\def\CC{{\mathbb C}} 
\def\DD{{\mathbb D}} 
\def\EE{{\mathbb E}}
\def\FF{{\mathbb F}}
\def\GG{{\mathbb G}}
\def\HH{{\mathbb H}}
\def\II{{\mathbb I}}
\def\JJ{{\mathbb J}}
\def\KK{{\mathbb K}}
\def\MM{{\mathbb M}}
\def\NN{{\mathbb N}}
\def\OO{{\mathbb O}}
\def\PP{{\mathbb P}}
\def\QQ{{\mathbb Q}}
\def\RR{{\mathbb R}}
\def\SS{{\mathbb S}}
\def\TT{{\mathbb T}}
\def\VV{{\mathbb V}}
\def\XX{{\mathbb X}}
\def\ZZ{{\mathbb Z}}
\def\kk{{\mathbbm k}}


%Italic
\def\cal{\mathcal}
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cE{{\cal E}}
\def\cF{{\cal F}}
\def\cG{{\cal G}}
\def\cH{{\cal H}}
\def\cI{{\cal I}}
\def\cJ{{\cal J}}
\def\cK{{\cal K}}
\def\cL{{\cal L}}
\def\cM{{\cal M}}
\def\cN{{\cal N}}
\def\cO{{\cal O}}
\def\cP{{\cal P}}
\def\cQ{{\cal Q}}
\def\cR{{\cal R}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
\def\cV{{\cal V}}
\def\cW{{\cal W}}
\def\cX{{\cal X}}
\def\cY{{\cal Y}}
\def\cZ{{\cal Z}}

%Greek
\def\a{\alpha}
\def\b{\beta}
\def\c{\gamma}
\def\d{\delta}
\def\e{\epsilon}
\def\g{\gamma}
\def\l{\lambda}
\def\L{\Lambda}
\def\o{\omega}
\def\s{\sigma}
\def\t{\tau}
\def\e{\varepsilon}
\renewcommand{\phi}{\varphi}

%emptyset
\def\vn{\varnothing}

%fat numbers
\def\ones{\mathbbm{1}}
\def\zeros{\mathbbm{0}}

%asterisk numbers
\def\xopt{x^*}
\def\yopt{y^*}
\def\Xopt{X^*}

% inverse and derivatives
\newcommand{\inv}{^{-1}}
\newcommand{\dl}{\partial}

% shorthand formatting
\newcommand{\til}{\widetilde}

%operator names 
\def\Aff{\operatorname {Aff}}
\def\conv{\operatorname {conv}}
\def\cone{\operatorname {cone}}
\def\cl{\operatorname {cl}}
\def\interior{\operatorname {int}}
\def\supp{\operatorname {supp}}
\DeclareMathOperator{\val}{val}
\DeclareMathOperator{\trop}{trop}
\def\Tr{\operatorname {Tr}}
\def\rk{\operatorname {rk}}
\def\Diag{\operatorname {Diag}}
\DeclareMathOperator{\diag}{diag}
\def\Vert{\operatorname {Vert}}
\def\diam{\operatorname {diam}}
\def\dist{\operatorname {dist}}
\def\spanset{\operatorname {span}}
\def\rowspan{\operatorname {rowspan}}
\def\Vol{\operatorname {Vol}}
\def\Var{\operatorname {Var}}
\def\Cov{\operatorname {Cov}}
\def\mean{\operatorname {mean}}
\def\med{\operatorname {median}}
\def\proj{\operatorname {proj}}
\def\Aut{\operatorname {Aut}}
\def\cut{\operatorname {CUT}_d^\square}

%lefts and rights
\def\lang{\left\langle}
\def\rang{\right\rangle}
\def\lnorm{\left\|}
\def\rnorm{\right\|}
\def\labs{\left|}
\def\rabs{\right|}
\def\lbbrace{\{\!\{}
\def\rbbrace{\}\!\}}
\newcommand{\evalat}[2]{\left.#1\right\rvert_{#2}}