On the volume of the Sp(<i>n</i>)⋅Sp(1) shadow of a compact set

Amedeo Altavilla; Lorenzo Nicolodi

doi:10.1016/j.crma.2015.12.007

Differential geometry

On the volume of the Sp(n)⋅Sp(1) shadow of a compact set

Presented by: the Editorial Board

Amedeo Altavilla ¹; Lorenzo Nicolodi ²

¹ Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
² Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 307-311.

Abstract
Résumé

Let $F : R^{4 n} \to R^{4 n}$ be an element of the quaternionic unitary group $Sp (n) \cdot Sp (1)$ , let K be a compact subset of $R^{4 n}$ , and let V be a $4 k$ -dimensional quaternionic subspace of $R^{4 n} ≅ H^{n}$ . The $4 k$ -dimensional shadow of the image under F of K is its orthogonal projection $P (F (K))$ onto V. We show that there exists a $4 k$ -dimensional quaternionic subspace W of $R^{4 n}$ such that the volume of the shadow $P (F (K))$ is the same as the volume of the section $K \cap W$ . This is a quaternionic analogue of the symplectic linear non-squeezing result recently obtained by Abbondandolo and Matveyev.

Soit $F : R^{4 n} \to R^{4 n}$ un élément du groupe unitaire quaternionnien $Sp (n) \cdot Sp (1)$ , soit K un ensemble compact dans $R^{4 n}$ , et soit V un sous-espace vectoriel quaternionnien de dimension $4 k$ dans $R^{4 n} ≅ H^{n}$ . L'ombre $4 k$ -dimensionnelle de l'image par F de K est sa projection orthogonale $P (F (K))$ sur V. Nous montrons qu'il existe un sous-espace vectoriel quaternionnien $W \subset R^{4 n}$ de dimension $4 k$ tel que le volume de l'ombre $P (F (K))$ est égal au volume de la section $K \cap W$ . Ceci est un analogue quaternionnien du résultat de non-squeezing lineaire symplectique obtenu récemment par Abbondandolo et Matveyev.

Received: 2015-09-30
Accepted: 2015-12-15
Published online: 2016-02-03

DOI: 10.1016/j.crma.2015.12.007

Author's affiliations:

Amedeo Altavilla ¹; Lorenzo Nicolodi ²

¹ Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
² Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

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     author = {Amedeo Altavilla and Lorenzo Nicolodi},
     title = {On the volume of the {Sp(\protect\emph{n})\ensuremath{\cdot}Sp(1)} shadow of a compact set},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {307--311},
     publisher = {Elsevier},
     volume = {354},
     number = {3},
     year = {2016},
     doi = {10.1016/j.crma.2015.12.007},
     language = {en},
}

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Amedeo Altavilla; Lorenzo Nicolodi. On the volume of the Sp(n)⋅Sp(1) shadow of a compact set. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 307-311. doi : 10.1016/j.crma.2015.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.007/

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Cited by Sources:

^☆ Authors partially supported by FIRB 2012 “Geometria differenziale e teoria geometrica delle funzioni” (A.A.); PRIN 2010–2011 “Varietà reali e complesse: geometria, topologia e analisi armonica” (L.N.); and by the GNSAGA of INdAM.

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