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<p>We provide an increasing approximating sequence of smooth lower bounds for the minimum of <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></EquationSource><EquationSource Format="TEX">$n\geq 2$</EquationSource></InlineEquation> 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 <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mi>n</mi><mo>=</mo><mn>2</mn></math></EquationSource><EquationSource Format="TEX">$n = 2$</EquationSource></InlineEquation>, where the solution relates to the well-known crossed-ladders problem.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月28日

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  • Thomas A. Weber

摘要

We provide an increasing approximating sequence of smooth lower bounds for the minimum of n≥2$n\geq 2$ 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.