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<p>In this paper we prove that the set of robustly quasiconvex functions is dense in the set of quasiconvex functions that obtain global minimum values on a closed bounded interval of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X=\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and a real-valued function defined on a convex set in a normed linear space <i>X</i> is quasiconvex iff the robustness radius of its epigraph is non-negative. In addition, a function is robustly quasiconvex if and only if its epigraph in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(X\times \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is robustly horizontally convex. A preservation property of convexity for robustly horizontally convex sets is established.</p>

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Density of Robustly Quasiconvex Functions in Quasiconvex Functions

  • 期刊论文

  • 发表日期：

    2026年2月13日

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  • Phan Thanh An

摘要

In this paper we prove that the set of robustly quasiconvex functions is dense in the set of quasiconvex functions that obtain global minimum values on a closed bounded interval of \(X=\mathbb {R}\) X = R and a real-valued function defined on a convex set in a normed linear space X is quasiconvex iff the robustness radius of its epigraph is non-negative. In addition, a function is robustly quasiconvex if and only if its epigraph in \(X\times \mathbb {R}\) X × R is robustly horizontally convex. A preservation property of convexity for robustly horizontally convex sets is established.