Smooth approximation of the minimum function from below (and above)

Abstract We provide an increasing approximating sequence of smooth lower bounds for the minimum of $n\geq 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> real numbers, with exponential convergence. For any given strict lower bound, we obtain a strict improvement through a harmonic increment. Additionally, we compute the maximum relative and absolute errors and construct a smooth approximating sequence from above using the geometric mean. Closed-form expressions are provided for the case $n = 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> , where the solution relates to the well-known crossed-ladders problem.