Let $x\in\mathbb R^d$ such that $||x||_2=1$. How to prove that the expected absolute value of a random walk on $x$ is at least $1/\sqrt 2$?

Question

Let $x\in\mathbb R^d$ such that $||x||_2=1$, and let $Y_1,\ldots,Y_d$ be i.i.d. random variables such that $Y_i=\begin{cases}1&\mbox{w.p. 1/2}\\-1&\mbox{otherwise}\end{cases}$.

Define $Z=\sum_{i=1}^d x_i\cdot Y_i$.

How can we show that $$\mathbb E[|Z|]\ge 1/\sqrt 2$$ for any such vector $x$?

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Notice that we have $\mathbb E[|Z|]= 1/\sqrt 2$ for the vector $x=(1/\sqrt 2, 1/\sqrt 2,0,\ldots,0)$.

Answer 1

This follows from Khintchine inequality.