5,282 research outputs found
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques
Double Bubbles Minimize
The classical isoperimetric inequality in R^3 states that the surface of
smallest area enclosing a given volume is a sphere. We show that the least area
surface enclosing two equal volumes is a double bubble, a surface made of two
pieces of round spheres separated by a flat disk, meeting along a single circle
at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as
described in the article. You can obtain this code by viewing the source of
this articl
Isoperimetric and stable sets for log-concave perturbations of Gaussian measures
Let be an open half-space or slab in endowed with
a perturbation of the Gaussian measure of the form
, where and is a smooth concave
function depending only on the signed distance from the linear hyperplane
parallel to . In this work we follow a variational approach to
show that half-spaces perpendicular to uniquely minimize the
weighted perimeter in among sets enclosing the same weighted volume.
The main ingredient of the proof is the characterization of half-spaces
parallel or perpendicular to as the unique stable sets with
small singular set and null weighted capacity. Our methods also apply for
, which produces in particular the classification of
stable sets in Gauss space and a new proof of the Gaussian isoperimetric
inequality. Finally, we use optimal transport to study the weighted minimizers
when the perturbation term is concave and possibly non-smooth.Comment: final version, to appear in Analysis and Geometry in Metric Space
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