Describing a Dual Cone 1)Does dual cone define just for proper cone or all kinds of cone ?
2)Can someone show me a figure that shows a dual cone of a cone ?
In Convex Optimization by Boyd (P.51) said that " $y\in k^*$ iff $-y$ is the normal of hyperplane that supports $k$ at the origin"
what does it mean? 

1) It is defined for all kinds of cone, even for an arbitrary set (see this link for more information).
2) A figure is shown in p. 143 of Dattorro, Jon. Convex optimization & Euclidean distance geometry. Meboo Publishing USA, 2005. For the explanation, see also pp.141-142. Fortunately, all of these pages are available in google books preview.
