SVM : support vector has margin of 0?

Question

I am trying to generate the separating hyperplane of binary classification for 3D points.

Here are my points, which are linearly separable.

Class 0: [[0,0,0], [0,1,1], [1,0,1], [0.5,0.4,0.4]]
Class 1: [[1,3,1], [2,0,2], [1,1,1]]

From sklearn.svm.SVC(kernel='linear'), the following is produced:

w = clf.coeff_ = [ 1.   0.5  0.5]
b = clf.intercept_ = -2.0
sv = clf.support_vectors_ = 
array([[ 0.,  1.,  1.],
       [ 1.,  0.,  1.],
       [ 2.,  0.,  2.],
       [ 1.,  1.,  1.]])

The understanding is, if w.dot(x)+b returns a negative value, then x is of Class 0; if positive value, then Class 1. However, w.dot([1,1,1])+b = 0 !! This means that [1,1,1], which is a support vector from Class 1, lies on the separating plane..... while no SVs from Class 0 lie on the sep. plane.

SO MY QUESTION IS...

My data is linearly separable, so theoretically an SVM should have margins >0 for both classes. But here, my SVM has a =0 for class1 and >0 margin for class0. Why is this the case? And if my hyperplane is incorrect, how can I calculate the correct hyperplane? Thank you.

CODE

from sklearn import svm
X0 = [[0,0,0], [0,1,1], [1,0,1], [0.5,0.4,0.4]]
Y0 = [0] * len(X0)
X1 = [[1,3,1], [2,0,2], [1,1,1]]
Y1 = [1] * len(X1)
X = X0 + X1
Y = Y0 + Y1
clf = svm.SVC(kernel='linear')
clf.fit(X, Y)
sv = clf.support_vectors_
w = clf.coef_[0] 
b = clf.intercept_[0]
print([w.dot(X0[i])+b for i in range(len(X0))]) # negative class
print([w.dot(X1[i])+b for i in range(len(X1))]) # positive class

Answer 1

By default, most SVM implementations are soft-margin SVM, which allows a point to be within the margin, or even on the wrong side of the decision boundary, even if the data is linearly separable. However, there is a penalty associated with each point which violates the traditional SVM constraints.

If we set the coefficient on the penalty to some very large value, we get something closer to hard-margin SVM which does what you'd expect:

>>> clf = svm.SVC(kernel='linear', C = 10000.)
>>> clf.fit(X, Y)
>>> w = clf.coef_[0] 
>>> b = clf.intercept_[0]
>>> print([w.dot(X0[i])+b for i in range(len(X0))]) # negative class
[-2.9999999999999987, -0.9999999999999987, -0.9999999999999991, -1.1999999999999988]
>>> print([w.dot(X1[i])+b for i in range(len(X1))]) # positive class
[5.000000000000002, 1.0000000000000004, 1.0000000000000009]

As you can see, the margins are now equal on both sides.