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if $ w_V(y) = |I_y|,\\text{ where } I_y := \\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\},$ then $$ \\lambda_1 \\geq \\frac{1}{250} \\min_{y > \\min V}{\\left(\\frac{1}{w_V(y)^2} + y\\right)}.$$ The result is sharp up to a universal constant if $\\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\}$ is an interval for the value of $y$ solving the 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