For a Minkowski centered convex compact set K we define <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha (K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to be the smallest possible factor to cover <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K \cap (-K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>∩</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by a rescalation of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\textbf {conv}}(K\cup (-K))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">conv</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>∪</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and give a complete description of the possible values of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha (K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the planar case in dependence of the Minkowski asymmetry of K. As a side product, we show that, if the asymmetry of K is greater than the golden ratio, the boundary of K intersects the boundary of its negative <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a result of Richter in 2018.

From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets

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发表日期：

2026年1月27日

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René Brandenberg,
Katherina von Dichter,
Bernardo González Merino

摘要

For a Minkowski centered convex compact set K we define $\alpha (K)$ $α (K)$ to be the smallest possible factor to cover $K \cap (-K)$ $K \cap (- K)$ by a rescalation of ${\textbf {conv}}(K\cup (-K))$ $conv (K \cup (- K))$ and give a complete description of the possible values of $\alpha (K)$ $α (K)$ in the planar case in dependence of the Minkowski asymmetry of K. As a side product, we show that, if the asymmetry of K is greater than the golden ratio, the boundary of K intersects the boundary of its negative $-K$ $- K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a result of Richter in 2018.