Right Quadruple Convexity of Complements
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> be a family of sets in <inline-formula><math xml...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-12-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/1/84 |
| _version_ | 1797625364634140672 |
|---|---|
| author | Xuemei He Liping Yuan Tudor Zamfirescu |
| author_facet | Xuemei He Liping Yuan Tudor Zamfirescu |
| author_sort | Xuemei He |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> be a family of sets in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula> (always <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>). A set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow></semantics></math></inline-formula> is called <i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-convex</i>, if for any pair of distinct points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>M</mi></mrow></semantics></math></inline-formula>, there is a set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>F</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⊂</mo><mi>M</mi></mrow></semantics></math></inline-formula>. A set of four points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>}</mo></mrow><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow></semantics></math></inline-formula> is called a <i>rectangular quadruple</i>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>conv</mi><mo>{</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>}</mo></mrow></semantics></math></inline-formula> is a non-degenerate rectangle. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> is the family of all rectangular quadruples, then we obtain the <i>right quadruple convexity</i>, abbreviated as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>q</mi></mrow></semantics></math></inline-formula>-<i>convexity</i>. In this paper we focus on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>q</mi></mrow></semantics></math></inline-formula>-convexity of complements, taken in most cases in balls or parallelepipeds. |
| first_indexed | 2024-03-11T09:55:31Z |
| format | Article |
| id | doaj.art-fc19ee3e2ccb41cba1e4e3a763463c2f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:55:31Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-fc19ee3e2ccb41cba1e4e3a763463c2f2023-11-16T15:53:11ZengMDPI AGMathematics2227-73902022-12-011118410.3390/math11010084Right Quadruple Convexity of ComplementsXuemei He0Liping Yuan1Tudor Zamfirescu2School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, ChinaSchool of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, ChinaSchool of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, ChinaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> be a family of sets in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></semantics></math></inline-formula> (always <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>). A set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow></semantics></math></inline-formula> is called <i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>-convex</i>, if for any pair of distinct points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>M</mi></mrow></semantics></math></inline-formula>, there is a set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>F</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⊂</mo><mi>M</mi></mrow></semantics></math></inline-formula>. A set of four points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>}</mo></mrow><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup></mrow></semantics></math></inline-formula> is called a <i>rectangular quadruple</i>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>conv</mi><mo>{</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>}</mo></mrow></semantics></math></inline-formula> is a non-degenerate rectangle. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> is the family of all rectangular quadruples, then we obtain the <i>right quadruple convexity</i>, abbreviated as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>q</mi></mrow></semantics></math></inline-formula>-<i>convexity</i>. In this paper we focus on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>q</mi></mrow></semantics></math></inline-formula>-convexity of complements, taken in most cases in balls or parallelepipeds.https://www.mdpi.com/2227-7390/11/1/84rectangular quadruplerq-convexitycomplements |
| spellingShingle | Xuemei He Liping Yuan Tudor Zamfirescu Right Quadruple Convexity of Complements Mathematics rectangular quadruple rq-convexity complements |
| title | Right Quadruple Convexity of Complements |
| title_full | Right Quadruple Convexity of Complements |
| title_fullStr | Right Quadruple Convexity of Complements |
| title_full_unstemmed | Right Quadruple Convexity of Complements |
| title_short | Right Quadruple Convexity of Complements |
| title_sort | right quadruple convexity of complements |
| topic | rectangular quadruple rq-convexity complements |
| url | https://www.mdpi.com/2227-7390/11/1/84 |
| work_keys_str_mv | AT xuemeihe rightquadrupleconvexityofcomplements AT lipingyuan rightquadrupleconvexityofcomplements AT tudorzamfirescu rightquadrupleconvexityofcomplements |