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<p>This dissertation studiess high dimensional problems from a low dimensional perspective.
First, we explore rectifiable curves in high-dimensional space by using the Fréchet
distance between and total curvatures of the two curves to bound the difference of
their lengths. We create this bound by mapping the curves into R^2 while preserving
the length between the curves and increasing neither</p><p>the total curvature of
the curves nor the Fr\'echet distance between them. The bound is independent of the
dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner
and Edelsbrunner for dimensions greater than three and it generalizes</p><p>a result
by F\'ary and Chakerian.</p><p>In the second half of the dissertation, we analyze
Gaussian mixtures. In particular, we consider the sum of n Gaussians, where each
Gaussian is centered at the vertex of a regular n-simplex. Fixing the width of the
Guassians and varying the diameter of the simplex from zero to infinity by increasing
a parameter that we call the scale factor, we find the window of scale factors for
which the Gaussian mixture has more modes, or local maxima, than components of the
mixture.</p><p>We see that the extra mode created is subtle, but can be higher than
the modes closer to the vertices of the simplex. In addition, we prove that all critical
points are located on a set of one-dimensional lines (axes) connecting barycenters
of complementary faces of</p><p>the simplex.</p>
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