Adding margin to a custom environment

Question

Is there a way to define a custom environment (for example Propositions) in such a way that the content of that environment will automatically have a given margin at the top and the bottom?

I mean something like

Here's a MWE:

% Document etc (fold)
        \documentclass[]{article}
        \usepackage[utf8]{inputenc}
        \usepackage[T1]{fontenc}
        \usepackage{amssymb,amsmath}
        \usepackage{amsmath}
        \usepackage{amsthm}
        \usepackage{mathrsfs}

        \newtheorem{prop}{Proposition}[]
 \newcommand{\calO}{\mathscr{O}}
        \newcommand{\OX}{\calO_X}
 \newcommand{\OC}{\calO_C}
 \newcommand{\OD}{\calO(D)}
 \newcommand{\OXD}{\OX(D)}
 \newcommand{\ND}{\calN_D}
 \newcommand{\ODD}{\calO_D(D)}
 \newcommand{\Mtt}{\tilde{M}_t}  
 \newcommand{\Xdr}{X_d^r}  
 \newcommand{\Xdrr}{ X_d^{\underline{r}} }  
 \newcommand{\Wdr}{W_d^r}  
 \newcommand{\Wdrr}{W_d^{\underline{r}}}  
 \newcommand{\Gdr}{G_d^r}  
 \newcommand{\eps}{\varepsilon}
 \newcommand{\AND}{\qquad \text{and} \qquad}
 \DeclareMathOperator{\Pic}{Pic}
 \newcommand{\ABiff}{if and only if }
 \newcommand{\al}{\alpha}
 \newcommand{\alb}{{\alpha\beta}}
 \newcommand{\set}[1]{\Big\{ #1 \Big\}}


    \begin{document}

            With the aim of describing the tangent space of $\Wdr$, we will first look at the one of its canonical blow-up $\Gdr$. A motivation for this approach is the observation that the natural projection
    $$ \pi:\Gdr\to\Wdr, \qquad (L,W) \mapsto L $$
    is biregular away of $W_d^{r+1}$. Indeed $\pi$ is clearly a regular map and, further, the preimage of a $L\in \Wdrr$ consists just of the point $w=(L,H^0(L))$. It follows that, as far as $\Wdrr$ is regarded, $\pi_*$ gives an isomorphism between the tangent spaces
    \begin{equation}\label{eq:tgnt_Wdrr}
            T_{w}\Gdr \cong T_L\Wdr, \quad \forall L\in\Wdrr.
    \end{equation}
    In order to describe the tangent space of $\Gdr$, a preliminary result about the first order deformations of a pair $(L,s) \in \Pic^d \times H^0(L)$ will turn out to be crucial.
    \begin{prop}
            Let $L\in \Pic^d$ be a line bundle over $X$ and $s\in H^0(L)$ a global section. Then an element $\phi \in T_L\Pic^d \cong H^0(\OX)$ induces a first order deformation of the pair $(L,s)$ \ABiff $\phi\cdot s=0$ in $H^1(L)$.
    \end{prop}
    \begin{proof}
            Assume that $L$ is given by transition functions $g_\alb$ on a open cover $U_\al$ of $X$. We already know that $T_L \Pic^d \cong H^1(\OX)$ and a first order deformation $L'$ of $L$ is represented by a class $\phi \in H^1(\OX)$ in the following way
            $$ g_\alb \quad\overset{\phi}\leadsto\quad g'_\alb = g_\alb \cdot (1+ \eps \phi_\alb). $$  
            On the other hand, on a first order deformation of the pair $(L,s)$ into $(L',s')$ we have the additional requirement that the section $s'$ corresponds to a linear deformation of $s$. In formula this is expressed as
            $$ s'_\al = s_\al + \eps t_\al, \quad t \in H^0(L). $$
            The action of the transition functions can therefore be expanded as
            $$ s'_\beta = g'_\alb \cdot s'_\al \;\iff\; s_\beta + \eps t_\beta = g_\alb \cdot (1 + \eps \phi_\alb)\cdot(s_\al + \eps t_\al)$$
            and imposes the conditions
            $$ s_\beta = g_\alb \cdot s_\al \AND \phi_\alb \cdot s_\al = t_\al - g_{\beta\al}\cdot t_\beta. $$
            The first one is automatically satisfied since $s$ being a global section of $L$, while the second one can be rewritten in terms of the coboundary map $\delta : C^0(L) \to C^1(L)$ as
            $$ \phi \cdot s = \delta (t), $$
            thus giving the desired result.
    \end{proof}

    \end{document}

Answer 1

You should define your own theorem style via \newtheoremstyle:

\documentclass{article}
\usepackage{amsthm}% http://ctan.org/pkg/amsthm
\usepackage[nopar]{lipsum}% http://ctan.org/pkg/lipsum

\newtheoremstyle{spacytheorem}
  {2\baselineskip} % Space above
  {2\baselineskip} % Space below
  {} % Body font
  {} % Indent amount
  {\bfseries} % Theorem head font
  {.} % Punctuation after theorem head
  {.5em} % Space after theorem head
  {} % Theorem head spec (can be left empty, meaning `normal')
\theoremstyle{spacytheorem}
\newtheorem{prop}{Proposition}

\begin{document}

\lipsum[1]
\begin{prop}
\lipsum[2]
\end{prop}
\begin{proof}
\lipsum[3]
\end{proof}
\lipsum[4]

\end{document}

The settings for \newtheoremstyle is presented in the amsthm documentation (section 4.3 New theorem styles, p 4).