A characterization of inner product spaces related to the p-angular distance

Dadipour, Farzad; Moslehian, Mohammad Sal

doi:10.1016/j.jmaa.2010.05.055

Cited by 18 publications

(16 citation statements)

References 17 publications

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“…where θ ̸ = kπ 2 , which in the particular case θ = π 4 coincides with the angle mentioned above, and based on it we present some new characterizations of inner product spaces. One can observe some analogies between these notions and the concept of p-angular distance and compare the related characterizations; see [2]. The spaces in this paper are all assumed to be real and X is often used to indicate a normed space.…”

Section: Introductionmentioning

confidence: 99%

Some characterizations of inner product spaces based on angle

Sales¹

2019

Hacettepe Journal of Mathematics and Statistics

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A problem in functional analysis that arises naturally is about finding necessary and sufficient conditions for a normed space to be an inner product space. By answering this question, mathematicians try to understand the inner product and normed spaces features. In this note, we have discussed this issue and we prove some results concerned with it. We introduce a notion of angle between two vectors in a normed space, denoted by A θ (., .) where θ ̸ = kπ 2 . We also speak about a notion of orthogonality concerning it, we call it θ-orthogonality. Mathematics Subject Classification (2010). 46C15

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Section: Introductionmentioning

confidence: 99%

Some characterizations of inner product spaces based on angle

Sales¹

2019

Hacettepe Journal of Mathematics and Statistics

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“…For some recently obtained upper and lower bounds for the p-angular distance the reader is referred to [8,9] and [16]. Numerous basic characterizations of inner product spaces under various conditions were first given by Fréchet, Jordan and von Neumann; see [4] and references therein. Since then, the problem of finding necessary and sufficient conditions for a normed space to be an inner product space has been investigated by many mathematicians by considering some types of orthogonality or some geometric aspects of underlying spaces; see, e.g., [11,15].…”

Section: Introductionmentioning

confidence: 99%

“…There is an interesting book by Amir [2] that contains several characterizations of inner product spaces, which are based on norm inequalities, various notions of orthogonality in normed linear spaces and so on. Among significant characterizations of inner product spaces related to p-angular distance, we can mention [1,4,5,6]. The next two theorems due to Lorch and Ficken will be used in this paper.…”

Section: Introductionmentioning

confidence: 99%

Geometric Aspects of $p$-angular and Skew $p$-angular Distances

Rooin¹,

Habibzadeh²,

Moslehian³

2018

Tokyo J. Math.

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Corresponding to the concept of p-angular distance α p [x, y] := x p−1 x − y p−1 y , we first introduce the notion of skew p-angular distance β p [x, y] := y p−1 x − x p−1 y for non-zero elements of x, y in a real normed linear space and study some of significant geometric properties of the p-angular and the skew p-angular distances. We then give some results comparing two different p-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the p-angular and the skew p-angular distances. In particular, we show that if p > 1 is a real number, then a real normed space X is an inner product space, if and only if for any2010 Mathematics Subject Classification. 46B20, 46C15, 26D15.

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“…There are several characterizations, and several of them were collected in the book of Amir [1] (see also e.g. [11,25,30,32] concerning some recent results). We emphasize the well-known Jordan-von Neumann theorem which was proven originally for complex spaces in [21], but as was pointed out there, real spaces can be handled along the same lines (even with some simplifications).…”

Section: Introductionmentioning

confidence: 99%

Is it possible to determine a point lying in a simplex if we know the distances from the vertices?

Gehér

2016

Journal of Mathematical Analysis and Applications

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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 |} d j=0 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 } d j=0 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 (Theorem 1) is re-proven using the fundamental theorem of projective geometry.2010 Mathematics Subject Classification. Primary: 46C15 Secondary: 15A63, 47A30.

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A characterization of inner product spaces related to the p-angular distance

Cited by 18 publications

References 17 publications

Some characterizations of inner product spaces based on angle

Some characterizations of inner product spaces based on angle

Geometric Aspects of $p$-angular and Skew $p$-angular Distances

Is it possible to determine a point lying in a simplex if we know the distances from the vertices?

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