What is a non-negative convex loss function? I'm strugling to understand this concept. In this book , it is explained that a non-negative convex loss function as: 

"...a decision space $D$ of elements $d$ and a non-negative convex
  loss function $L(d,\theta)$. For example, in connection with the normal model described at the end of the previous paragraph, $L(d,\theta)$ might be the following "squared error" loss function, $L(d,\mu)= c(\mu- d)^2$, where $c$ is a given positive constant and $d = d(x)$ is some estimate of $\mu$ belonging to a decision space $D$. The problem is then to choose a $d$ from $D$ that is in some sense optimal relative to the loss function $L(d,\mu)$."

I understand that non-negative convex derives from the fact that it is a quadratic function, with no negative values in the range, which is convex. However, why it is a loss function? and what do they mean by a decision space? will $d=d(x)$ represents all the elements in the sample space? what is the role of the constant $c$? 
Thanks a lot.
 A: Let $\theta$ denote an unknown quantity that affects the decision process. We call $\theta$ the state of the world. Let $\Theta$ denote the parameter space, such that $\Theta=\{ \text{the state of all possible states of nature}\}$. Let $d$ denote the decision that might change depending on $\theta$, and $D$ be the set of all possible decisions.
Given that particular action $d$ is taken and $\theta$ in fact is true state of the world then one can consider a loss function (utility function) $L(\theta, d)$. It is important to note, that $L(\theta, \theta)$ = 0. In statistical problems,
$\theta$ is a often an unknown parameter that needs to be estimated, and the decision $d$ corresponds to a point estimator.
The $L(d,\mu)$ corresponds to squared-error loss and the $c$ can be thought of as a scaled version that penalizes you more when $c>1$.
You should pick up any book on Decision Theory if you really want to read/know more though about loss functions. Although, if you are comfortable with Bayesian statistics, I highly recommend this book for a good explanation of loss functions: The Bayesian Choice
