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We show that there exists a fixed unit vector $\\theta \\in \\mathbb{R}^n$ such that the random variable $Y = \\langle X, \\theta \\rangle$ satisfies $$ \\min \\left \\{ \\mathbb{P}( Y \\geq t M ), \\mathbb{P}(Y \\leq -tM) \\right \\} \\geq c e^{-C t^2} \\qquad \\qquad \\text{for all} \\ 0 \\leq t \\leq \\tilde{c} \\sqrt{n}, $$ where $M > 0$ is any median of $|Y|$, i.e., $\\min \\{ \\mathbb{P}( |Y| \\geq M), \\mathbb{P}( |Y| \\leq M ) \\} \\geq 1/2$. 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The density function can be arbitrary. We show that there exists a fixed unit vector $\\theta \\in \\mathbb{R}^n$ such that the random variable $Y = \\langle X, \\theta \\rangle$ satisfies $$ \\min \\left \\{ \\mathbb{P}( Y \\geq t M ), \\mathbb{P}(Y \\leq -tM) \\right \\} \\geq c e^{-C t^2} \\qquad \\qquad \\text{for all} \\ 0 \\leq t \\leq \\tilde{c} \\sqrt{n}, $$ where $M > 0$ is any median of $|Y|$, i.e., $\\min \\{ \\mathbb{P}( |Y| \\geq M), \\mathbb{P}( |Y| \\leq M ) \\} \\geq 1/2$. 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