Affine linear classifier ($h(x)=1_{\{w\cdot x+b\ge 0\}}$)

An affine linear classifier assigns binary labels according to the sign of an affine function. Its boundary is an affine hyperplane, and its positive region is a closed half-space under the displayed convention. On ${\mathbb{R}}^d$ the full class has VC dimension $d+1$: arbitrary signs interpolate on $d+1$ affinely independent points, while the intersecting convex hull partition theorem prevents shattering any $d+2$ points. Requiring $b=0$ gives the homogeneous case and a different VC dimension.