v2004.06.19 - Convex Optimization

2.3. HALFSPACE, HYPERPLANE 37H +ay pcyd∆∂H={y | a T y=b}H −N(a T )={y | a T y=0}Figure 2.4: Hyperplane illustrated ∂H is a line partially bounding halfspacesH − = {y |a T y ≤ b} and H + = {y |a T y ≥ b} in R 2 . Shaded is a rectangularpiece of semi-infinite H − with respect to which vector a is outward-normalto bounding hyperplane; vector a is inward-normal with respect to H + . Inthis particular instance, H − contains nullspace N(a T ) (dashed line throughorigin) because b > 0. Hyperplane, halfspace, and nullspace are each drawntruncated. Points c and d are equidistant from hyperplane, and vectorc − d is normal to it. ∆ is distance from origin to hyperplane.