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A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game. (English) Zbl 1127.26006
In a recent paper [Rev. Mat. Iberoam. 21, 889–910 (2005; Zbl 1116.26007)] Z. Buczolich solved Weil’s gradient problem by establishing the existence of a differentiable function $$f : {\mathbb R}^2 \to {\mathbb R}$$ such that $${\nabla}f(0) = 0$$ and $$| {\nabla}f | \geq 1$$ almost everywhere. In this paper the authors give a simpler construction of such a function. For this purpose they consider the following infinite game.
Let $$B = B(0,R)$$ be an open disk in $${\mathbb R}^2$$. The point-line game PL is a sequence of rounds. The first and the second player choose points $${\boldsymbol a}_k$$ in $$B$$ and lines $$p_k$$, respectively, obeying the following rules. In the first round, the first player chooses a point $${\boldsymbol a}_1 \in B$$ and then the second player chooses a line $$p_1$$ with $${\boldsymbol a}_1 \in p_1$$. In the $$k$$-th round, where $$k > 1 \,$$, the first player chooses a point $${\boldsymbol a}_k \in B \cap p_{k-1}$$ and the second player chooses a line $$p_k$$ passing through $${\boldsymbol a}_k \,$$. The first player wins if the sequence $$({\boldsymbol a}_k)$$ diverges, otherwise the second player wins.
Two proofs are given for the crucial statement that the second player has a winning strategy in the game PL. Based on this result, a new proof of Buczolich’s theorem is presented in the paper.

##### MSC:
 26B05 Continuity and differentiation questions 91A20 Multistage and repeated games 91A80 Applications of game theory
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