Linear Discriminant (decision region) singly connected and convex? I am reading about linear discriminants, and have encountered a phrase that I have no idea about. The phrase says that a decision region constructed in a certain way, is "singly connected and convex", I want to know in really simple terms what this means. I have an idea of what convex means in that a convex function has a definite optimum. But I don't know what singly connected means and I'm not sure if convex means the same thing here.
This question might be very simple but I am very new to the field of statistical learning and am struggling with the concepts related to optimization theory so any help would be much appreciated.
Thanks in advance! 
 A: I suggest you read Pattern Classification by Duda and Hart.
Quote from wikipedia:
"In Euclidean space, a region is convex if for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex." Your notion of a convex function would be the boundary of a convex set. See the wiki page for examples, they are pretty clear.
You can show that the region is convex, by showing that for two points $x_1$ and $x_2$ in a decision region $R_1$, $x_3=(a \cdot x_1 + (1-a)\cdot x_2)$ is also in the decision region $R_1$ (for some variable $a \in (0,1)$ ).
Quote from Pattern Classification, by Duda and Hart:
"In particular, for good performance every decision region should be singly connected, and this tends to make the linear machine most suitable for problems for which the conditional densities $p(x\mid \omega_i)$ are unimodal." Meaning that while any convex region is simply connected, the classifier works best if decision regions don't have multiple overlaps.
