Hyperplan

Résumé

En mathématiques et plus particulièrement en algèbre linéaire et géométrie, les hyperplans d'un espace vectoriel E de dimension quelconque sont la généralisation des plans vectoriels d'un espace de dimension 3 : ce sont les sous-espaces vectoriels de codimension 1 dans E. Si E est de dimension finie n non nulle, ses hyperplans sont donc ses sous-espaces de dimension n – 1 : par exemple l'espace nul dans une droite vectorielle, une droite vectorielle dans un plan vectoriel Caractérisation Soient E un espace vectoriel et H un sous-espace. Les propositions suivantes sont équivalentes : #H est un hyperplan de E ; #il existe dans E une droite vectorielle supplémentaire de H ; #H ≠ E et toute droite vectorielle de E engendrée par un vecteur n'appartenant pas à H est un supplémentaire de H ; #H est le noyau d'une forme linéaire non nulle ; #H est défini par une équation linéaire homogène non triviale. Exemples

Dans le K-espace vectoriel des matrices carrées d'ordre

Source officielle

https://fr.wikipedia.org/wiki/Hyperplan

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Séances de cours associées (29)

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New Moduli For Banach Spaces

Grigory Ivanov

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, for example, as lengths of catheti of right-angled triangles (defined via so-called quasiorthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things, it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander-type results involving generalizations of the modulus of convexity and the modulus of Banas.

Tusi Mathematical Research Group2017

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Given a set V of points in , two points p, q from V form a double-normal pair, if the set V lies between two parallel hyperplanes that pass through p and q, respectively, and that are orthogonal to the segment pq. In this paper we study the maximum number of double-normal pairs in a set of n points in . It is not difficult to get from the famous ErdAs-Stone theorem that for a suitable integer and it was shown in a paper by Pach and Swanepoel that and that asymptotically . In this paper we sharpen the upper bound on k(d), which, in particular, gives and in addition to the equality established by Pach and Swanepoel. Asymptotically we get and show that this problem is connected with the problem of determining the maximum number of points in that form only acute (or non-obtuse) angles.

Springer2016

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A slab (or plank) is the part of the d-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the width of the slab. It is conjectured that the members of any infinite family of slabs with divergent total width can be translated so that the translates together cover the whole d-dimensional space. We prove a slightly weaker version of this conjecture, which can be regarded as a converse of Bang's theorem, also known as Tarski's plank problem. This result enables us to settle an old conjecture of Makai and Pach on simultaneous approximation of polynomials. We say that an infinite sequence S of positive numbers controls all polynomials of degree at most d if there exists a sequence of points in the plane whose x-coordinates form the sequence S, such that the graph of every polynomial of degree at most d passes within vertical distance 1 from at least one of the points. We prove that a sequence S has this property if and only if the sum of the reciprocals of the dth powers of its elements is divergent.

Mathematical Assoc Amer2017

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