Largest parallelotopes contained in simplices

AbstractWe establish in this paper a theorem for the volume of the largest parallelotope contained in a given simplex. Applying this theorem, we prove some inequalities for unions of parallelotopes in a given simplex and some spanning theorems for inscribed simplices

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.Comment: 15 page

Least squares approximations of measures via geometric condition numbers

For a probability measure on a real separable Hilbert space, we are interested in "volume-based" approximations of the d-dimensional least squares error of it, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of that measure, where the comparison depends on a simple quantitative geometric property of it. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of measure. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo SVD based on volume sampling