v2007.11.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 61H +ay pcyd∆∂H={y | a T (y − y p )=0}H −N(a T )={y | a T y=0}Figure 18: Hyperplane illustrated ∂H is a line partially bounding halfspacesH − = {y | a T (y − y p )≤0} and H + = {y | a T (y − y p )≥0} in R 2 . Shaded isa rectangular piece of semi-infinite H − with respect to which vector a isoutward-normal to bounding hyperplane; vector a is inward-normal withrespect to H + . Halfspace H − contains nullspace N(a T ) (dashed linethrough origin) because a T y p > 0. Hyperplane, halfspace, and nullspace areeach drawn truncated. Points c and d are equidistant from hyperplane, andvector c − d is normal to it. ∆ is distance from origin to hyperplane.