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Let $X, Y \subset \mathbb{Z}^2$ be two discrete and bounded sets. Let $f_X$ be the Euclidean signed distance function of $X$ (similarly for $Y$) and $d_H(X,Y)$ the Euclidean Haussdorff distance between them. Then, is it true that $$ |f_X(z) - f_Y(z)| \leq d_H(X,Y) $$ for each $z \in \mathbb{Z}^2$?

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  • $\begingroup$ Yes it is true, in fact the optimal upper bound of the left hand side some times used as a definition of Hausdorff distance. See for example "Kreis Und Kugel" by Blaschke. $\endgroup$
    – Anton Petrunin
    Commented Nov 4, 2015 at 10:09

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It is not true. The correct bound can be found in Theorem 2 (p. 5) of this paper.

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