Is it possible to determine a point lying in a simplex if we know the distances from the vertices?
It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivate...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Gehér, György Pál [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016transfer abstract |
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| Schlagwörter: |
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| Umfang: |
13 |
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| Übergeordnetes Werk: |
Enthalten in: In silico drug repurposing in COVID-19: A network-based analysis - Sibilio, Pasquale ELSEVIER, 2021, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:439 ; year:2016 ; number:2 ; day:15 ; month:07 ; pages:651-663 ; extent:13 |
| Links: |
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| DOI / URN: |
10.1016/j.jmaa.2016.03.024 |
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| 520 | |a It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. | ||
| 520 | |a It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. | ||
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10.1016/j.jmaa.2016.03.024 doi GBVA2016022000019.pica (DE-627)ELV035608072 (ELSEVIER)S0022-247X(16)00242-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 44.40 bkl Gehér, György Pál verfasserin aut Is it possible to determine a point lying in a simplex if we know the distances from the vertices? 2016transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. Resolving set Elsevier Simplex Elsevier Bisector Elsevier Enthalten in Elsevier Sibilio, Pasquale ELSEVIER In silico drug repurposing in COVID-19: A network-based analysis 2021 Amsterdam [u.a.] (DE-627)ELV006634001 volume:439 year:2016 number:2 day:15 month:07 pages:651-663 extent:13 https://doi.org/10.1016/j.jmaa.2016.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-PHA 44.40 Pharmazie Pharmazeutika VZ AR 439 2016 2 15 0715 651-663 13 045F 510 |
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10.1016/j.jmaa.2016.03.024 doi GBVA2016022000019.pica (DE-627)ELV035608072 (ELSEVIER)S0022-247X(16)00242-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 44.40 bkl Gehér, György Pál verfasserin aut Is it possible to determine a point lying in a simplex if we know the distances from the vertices? 2016transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. Resolving set Elsevier Simplex Elsevier Bisector Elsevier Enthalten in Elsevier Sibilio, Pasquale ELSEVIER In silico drug repurposing in COVID-19: A network-based analysis 2021 Amsterdam [u.a.] (DE-627)ELV006634001 volume:439 year:2016 number:2 day:15 month:07 pages:651-663 extent:13 https://doi.org/10.1016/j.jmaa.2016.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-PHA 44.40 Pharmazie Pharmazeutika VZ AR 439 2016 2 15 0715 651-663 13 045F 510 |
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10.1016/j.jmaa.2016.03.024 doi GBVA2016022000019.pica (DE-627)ELV035608072 (ELSEVIER)S0022-247X(16)00242-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 44.40 bkl Gehér, György Pál verfasserin aut Is it possible to determine a point lying in a simplex if we know the distances from the vertices? 2016transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. Resolving set Elsevier Simplex Elsevier Bisector Elsevier Enthalten in Elsevier Sibilio, Pasquale ELSEVIER In silico drug repurposing in COVID-19: A network-based analysis 2021 Amsterdam [u.a.] (DE-627)ELV006634001 volume:439 year:2016 number:2 day:15 month:07 pages:651-663 extent:13 https://doi.org/10.1016/j.jmaa.2016.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-PHA 44.40 Pharmazie Pharmazeutika VZ AR 439 2016 2 15 0715 651-663 13 045F 510 |
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10.1016/j.jmaa.2016.03.024 doi GBVA2016022000019.pica (DE-627)ELV035608072 (ELSEVIER)S0022-247X(16)00242-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 44.40 bkl Gehér, György Pál verfasserin aut Is it possible to determine a point lying in a simplex if we know the distances from the vertices? 2016transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. Resolving set Elsevier Simplex Elsevier Bisector Elsevier Enthalten in Elsevier Sibilio, Pasquale ELSEVIER In silico drug repurposing in COVID-19: A network-based analysis 2021 Amsterdam [u.a.] (DE-627)ELV006634001 volume:439 year:2016 number:2 day:15 month:07 pages:651-663 extent:13 https://doi.org/10.1016/j.jmaa.2016.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-PHA 44.40 Pharmazie Pharmazeutika VZ AR 439 2016 2 15 0715 651-663 13 045F 510 |
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10.1016/j.jmaa.2016.03.024 doi GBVA2016022000019.pica (DE-627)ELV035608072 (ELSEVIER)S0022-247X(16)00242-0 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 44.40 bkl Gehér, György Pál verfasserin aut Is it possible to determine a point lying in a simplex if we know the distances from the vertices? 2016transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. Resolving set Elsevier Simplex Elsevier Bisector Elsevier Enthalten in Elsevier Sibilio, Pasquale ELSEVIER In silico drug repurposing in COVID-19: A network-based analysis 2021 Amsterdam [u.a.] (DE-627)ELV006634001 volume:439 year:2016 number:2 day:15 month:07 pages:651-663 extent:13 https://doi.org/10.1016/j.jmaa.2016.03.024 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA SSG-OPC-PHA 44.40 Pharmazie Pharmazeutika VZ AR 439 2016 2 15 0715 651-663 13 045F 510 |
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is it possible to determine a point lying in a simplex if we know the distances from the vertices? |
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Is it possible to determine a point lying in a simplex if we know the distances from the vertices? |
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It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. |
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It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. |
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It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these () is re-proven using the fundamental theorem of projective geometry. |
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