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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2025-01-01
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| author | Xinling Zhang |
| author_facet | Xinling Zhang |
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| description | 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. |
| format | Article |
| id | doaj-art-a75c00540b374b959b0397b7a72f9548 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-a75c00540b374b959b0397b7a72f95482025-01-24T13:22:15ZengMDPI AGAxioms2075-16802025-01-011414810.3390/axioms14010048Partitioning Functional of a Class of Convex BodiesXinling Zhang0School of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaFor 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.https://www.mdpi.com/2075-1680/14/1/48Borsuk’s problemconvex bodiespartitioning functional |
| spellingShingle | Xinling Zhang Partitioning Functional of a Class of Convex Bodies Axioms Borsuk’s problem convex bodies partitioning functional |
| title | Partitioning Functional of a Class of Convex Bodies |
| title_full | Partitioning Functional of a Class of Convex Bodies |
| title_fullStr | Partitioning Functional of a Class of Convex Bodies |
| title_full_unstemmed | Partitioning Functional of a Class of Convex Bodies |
| title_short | Partitioning Functional of a Class of Convex Bodies |
| title_sort | partitioning functional of a class of convex bodies |
| topic | Borsuk’s problem convex bodies partitioning functional |
| url | https://www.mdpi.com/2075-1680/14/1/48 |
| work_keys_str_mv | AT xinlingzhang partitioningfunctionalofaclassofconvexbodies |