v2009.01.01 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 67
H +
a
y p
c
y
d
∂H = {y | a T (y − y p )=0} = N(a T ) + y p
∆
H −
N(a T )={y | a T y=0}
Figure 21: Hyperplane illustrated ∂H is a line partially bounding halfspaces
H − = {y | a T (y − y p )≤0} and H + = {y | a T (y − y p )≥0} in R 2 . Shaded is
a rectangular piece of semi-infinite H − with respect to which vector a is
outward-normal to bounding hyperplane; vector a is inward-normal with
respect to H + . Halfspace H − contains nullspace N(a T ) (dashed line
through origin) because a T y p > 0. Hyperplane, halfspace, and nullspace are
each drawn truncated. Points c and d are equidistant from hyperplane, and
vector c − d is normal to it. ∆ is distance from origin to hyperplane.