Convex optimization in svm

So the SVM constraints are actually linear in the unknowns

Now any linear constraint defines a convex set and a set of simultaneous linear constraints defines the intersection of convex sets, so it is also a convex set

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When using the Euclidian norm, the SVM objective function (besided being convex) is quadratic because the Euclidian norm is equivalent to an inner product of w with itself

Finally, the usual definition o a convex problem is optimize a convex function over a convex set