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Suppose that

\displaystyle{{v_1},{v_2},...,{v_d}}

are unit vectors in

\displaystyle{{{\Bbb R}^d}}

. Prove that there exists a unitary vector

\displaystyle{u}

such that

\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}

for

\displaystyle{i = 1,2,...,d}

.
Note. Here

\displaystyle{ \cdot }

denotes the usual scalar product on

\displaystyle{{{\Bbb R}^d}}

.
Proposed by Tomasz Tkocz, University of Warwick.

Problem Statement