points for which li these pointswhich are also support vectorslie strictly inside the margins (points and in the figuresuch points may or may not be correctly classified points for which li these are the non-support vectorswhich all lie outside the margins every such point is correctly classified if the classes of points {xi yi and {xi yi - are perfectly separable by some affine planethen there will be no points strictly inside the marginsso all support vectors will lie exactly on the margins in this case ( reduces to min kbk , subject toyi ( > ( using the fact that and ka xxa xb we may replace min kbk in ( with max /kbkas this gives the same optimal solution as /kbk is equal to half the margin widththe latter optimization problem has simple interpretationseparate the points via an affine hyperplane such that the margin width is maximized example (support vector machinethe data in figure was uniformly generated on the unit disc class- points (blue dotshave radius less than / ( -values and class- points (red crosseshave radius greater than / ( -values - - - - - - - figure separate the two classes of course it is not possible to separate the two groups of points via straight line in howeverit is possible to separate them in by considering three-dimensional feature vectors [ ][ ]for any the corresponding feature vector lies on quadratic surface in this space it is possible to separate the {zi points into two groups by means of planar surfaceas illustrated in figure