Optimizing absolute Gaussian curvature locally

Given a set of points in a Hilbert space that can be separated from
the origin. The slab support vector machine (slab SVM) is an
optimization problem that aims at finding a slab (two parallel
hyperplanes whose distance---the slab width---is essentially fixed)
that encloses the points and is maximally separated from the origin.
Extreme cases of the slab SVM include the smallest enclosing ball
problem and an interpolation problem that was used (as the slab SVM
itself) in surface reconstruction with radial basis functions. Here
we show that the path of solutions of the slab SVM, i.e., the
solution parameterized by the slab width is piecewise linear and can
be computed efficiently.

We show that given a polygon $C$ in $\mathbb{R}^3$ it is
\emph{algebraically} hard to find a point in $\mathbb{R}\setminus C$
at which the absolute curvature with respect to $C$ is minimized. By
algebraically hard we mean that in general an optimal solution is
not constructible, i.e., there exist no finite sequences of
expressions starting with a rational number, where each expression
is either the sum, difference, product, quotient or $k$'th root of
preceding expressions and the last expressions give the coordinates
of a sought for optimal solution. We also design an approximation
scheme for the value of the curvature at an optimal solution.