Prove that, for any unit vectors \( v_1, v_2, \cdots, v_n \) in \( \mathbb{R}^n \), there exists a unit vector \( w \) in \( \mathbb{R}^n \) such that \( \langle w, v_i \rangle \leq n^{-1/2} \) for all \( i = 1, 2, \cdots, n \). (Here, \( \langle \cdot, \cdot \rangle \) is a usual scalar product in \( \mathbb{R}^n \).)
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2014-13 Unit vectors,
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well, it seems late, but is it correct that problem asks about absolute value of dot product?
The statement in the problem holds with or without the absolute value on the dot product.
Of course it should hold with or without the absolute value, but without the absolute value, it is not so hard to find a unit vector that can make all the dot products nonpositive, which i guess is slightly off the original intention.
While I can see your point, the problem as it is does not seem to be absurdly easy, so I guess it is not a big issue here. (Moreover, in the same spirit, one can claim that the problem with the absolute value is not so hard either.)