The aim of this paper is to motivate the development of a Brunn-Minkowski
theory for minimal surfaces. In 1988, H. Rosenberg and E. Toubiana
studied
a
sum operation for finite total curvature complete minimal surfaces in
$\mathbb{R}^{3}$ and noticed that minimal hedgehogs of $\mathbb{R}^{3}
$ constitute a real vector space [14]. In 1996, the author
noticed
that
the square root of the area of minimal hedgehogs of $\mathbb{R}^{3}$
that
are
modelled on the closure of a connected open subset of
$\mathbb{S}^{2}$ is
a
convex function of the support function [5]. In this paper, the
author
¶ (i) gives new geometric inequalities for minimal
surfaces
of
$\mathbb{R}^{3}$;
¶ (ii) studies the relation between
support
functions and Enneper-Weierstrass representations;
¶ (iii)
introduces and studies a new type of addition for minimal surfaces;
¶ (iv) extends notions and techniques from the classical
Brunn-Minkowski theory to minimal surfaces. Two characterizations of the
catenoid among minimal hedgehogs are given.