Partitioning Functional of a Class of Convex Bodies
For each <i>n</i>-dimensional real Banach space <i>X</i>, each positive integer <i>m</i>, and each bounded set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/48 |
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| Summary: | For each <i>n</i>-dimensional real Banach space <i>X</i>, each positive integer <i>m</i>, and each bounded set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>X</mi></mrow></semantics></math></inline-formula> with diameter greater than 0, let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mi>X</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be the infimum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>X</mi></mrow></semantics></math></inline-formula> can be represented as the union of <i>m</i> subsets of <i>A</i>, whose diameters are not greater than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> times the diameter of <i>A</i>. Estimating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mi>X</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is an important part of Chuanming Zong’s quantitative program for attacking Borsuk’s problem. However, estimating the partitioning functionals of general convex bodies in finite dimensional Banach spaces is challenging, so we will begin with the estimation of partitioning functionals for special convex bodies. In this paper, we prove a series of inequalities about partitioning functionals of convex cones. Several estimations of partitioning functionals of the convex hull of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>+</mo><mi>u</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mi>A</mi><mo>−</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>+</mo><mi>u</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mo>−</mo><mi>A</mi><mo>−</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> are also presented, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⊆</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> is a convex body with the origin <i>o</i> in its interior, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mrow><mo>∖</mo></mrow><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. These results contribute to the study of Borsuk’s problem through Zong’s program. |
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| ISSN: | 2075-1680 |