August 20, 2025

The Dyn–Farkhi conjecture and the convex hull of a sumset in two dimensions

Key Points

The research resolves the Dyn–Farkhi conjecture for compact sets when n equals 2, confirming essential geometric properties.
Key evidence includes proving a new representation of sumsets in two dimensions, enhancing mathematical understanding of distances.
This paper employs techniques related to the Hausdorff metric and convex hulls to derive these findings within a geometric framework.
The significance of this work lies in its potential applications in optimizing geometric structures and distance calculations in multidimensional spaces.

Abstract

We study the Hausdorff distance to convex hull, which for a compact set A ⊂ R n A Rⁿ is defined by d (A) ≔ d H (A, c o n v (A) ) d (A) ≔dH (A, conv (A) ), where d H dH is the Hausdorff metric. In 2004, Dyn and Farkhi Numer. Funct. Anal. Optim. 25 (2004), pp. 363–377 conjectured that d 2 d² is subadditive on compact sets in R n 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 n 3. In this paper, we resolve the Dyn–Farkhi conjecture when n = 2 n=2. In doing so, we prove a new representation of the sumset c o n v (A) + c o n v (B) conv (A) + conv (B) for compact sets A, B ⊂ R 2 A, B R².

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