# SVMs and separating margins

A couple of points that I need cleared up, thanks ahead for any help.

• In a D dimensional SVM what is the dimension of the maximal separating plane? Intuitively (and following VC logic) it will be D-1, is that correct....
• In the derivation of the SVM, what roles do the Lagrangian multiplier play? What do their values indicate?
• It is a matter of some topological interest that no subspace (or even differentiable submanifold) of dimension less than $D-1$ can separate $D$ dimensional space into two or more connected components. Of greater import is your second question. You might enjoy reading some of our higher-voted threads on SVMs, such as stats.stackexchange.com/questions/3947.
– whuber
Apr 28, 2014 at 22:47

The SVM separating hyperplane is defined as the set of all points $x$ satisfying $w^t x - b = 0$ where $w$ is some nonzero vector (it is the normal to the separating hyperplane).
It is easy to see that $\{x : w^t x - b = 0\}$ is an affine space of dimension $D-1$ since it is equal to $b + w^{\perp}$ and $\dim w^\perp = D-1$ where $w^{\perp}$ denotes the space perpendicular to the span of the vector $w$.