Decreasing paths of polygons, (with Scott Whitman, Isaac Kulp, Dr. Leonel Robert Gonzalez, and Dr. Charlotte Ochanine), Discrete & Computational Geometry. 2025.

Journal: https://doi.org/10.1007/s00454-025-00750-5

ArXiv: https://doi.org/10.48550/arXiv.2402.12643

Abstract: We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by a decreasing path (attainability problem), and we show that this can be done by a finite application of “pull-in” moves (bang–bang problem). In the case of triangles, these problems were investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.

Building homotheties through pinches, (with Franziska Riepl, Dr. George Turcu, Dr. Leonel Robert Gonzalez, and Lucas Walls), Linear and Multilinear Algebra. 2021. 

Journal: https://doi.org/10.1080/03081087.2021.1993124

PDF: https://userweb.ucs.louisiana.edu/~C00254569/Pinches2.pdf

Abstract: We show that starting from finitely many points in \R^m, and successively applying ‘pinches’ on them, it is possible to arrive at any of the configurations that result from applying, to the original points, a homothety of factor 0 \leq s \leq 1 and centre the centroid of the points. Here, a ‘pinch’ consists in moving two points towards their common centroid. For three points, four pinches suffice. In general, the number of pinches is independent of the original configuration and of m.