A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization

In a recent issue of this journal, Mordukhovich et al.\ pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem one is given $k+1$ closed convex sets in \Real^d equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first $k$ sets is minimal. In later work the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the convex case. Here, we revisit the original problem solely from the numerical perspective. By exploiting the majorization-minimization (MM) principle of computational statistics and rudimentary techniques from differential calculus, we are able to construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem.Comment: 21 pages, 3 figure

About the continuity of one operation with convex compacts in finite-dimensional normed spaces

In this paper, we study the deformation of the intersection of one compact set with a closed neighborhood of another compact set by changing the radius of this neighborhood. It is shown that in finite-dimensional normed spaces, in the case when both compact sets are non-empty convex subsets, such an operation is continuous in the topology generated by the Hausdorff metric. The question of the continuous dependence of the described intersection on the radius of the neighborhood arose as a by-product of the development of the theory of extremal networks. However, it turned out to be interesting in itself, suggesting various generalizations. Therefore, it was decided to publish it separately.Comment: in Russian languag

On the Optimality of Napoleon Triangles

An elementary geometric construction, known as Napoleon’s theorem, produces an equilateral triangle, obtained from equilateral triangles erected on the sides of any initial triangle: The centers of the three equilateral triangles erected on the sides of the arbitrarily given original triangle, all outward or all inward, are the vertices of the new equilateral triangle. In this note, we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle, whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle, whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances.For more information: Kod*la