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

We provide an increasing approximating sequence of smooth lower bounds for the minimum of n≥ 2 n ≥ 2 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 $n = 2$ n = 2 , where the solution relates to the well-known crossed-ladders problem.