The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions
No Thumbnail Available
Authors
Meyer, Mark
Advisors
Issue Date
2025-20-08
Type
Article
Keywords
Convex hull , Hausdorff metric , Minkowski sum , Subadditive.
Citation
Meyer, M. (2025). The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions. Proceedings of the American Mathematical Society, 153(11). https://doi.org/10.1090/proc/17313
Abstract
We study the Hausdorff distance to convex hull, which for a compact set A ? Rn is defined by d(A):= dH(A, conv(A)), where dH is the Hausdorff metric. In 2004, Dyn and Farkhi [Numer. Funct. Anal. Optim. 25 (2004), pp. 363-377] conjectured that d2 is subadditive on compact sets in Rn. In 2018, Fradelizi, Madiman, Marsiglietti, and Zvavitch [EMS Surv. Math. Sci. 5 (2018), pp. 1-64] found a counterexample to this conjecture when n ? 3. In this paper, we resolve the Dyn-Farkhi conjecture when n = 2. In doing so, we prove a new representation of the sumset conv(A)+conv(B) for compact sets A,B ? R2.
Table of Contents
Description
Click on the DOI link to access this article at the publishers website (may not be free).
Publisher
American Mathematical Society
Journal
Proceedings of the American Mathematical Society
Book Title
Series
PubMed ID
ISSN
1088-6826
0002-9939
0002-9939