The space of isometry covariant tensor valuations

Hug, D.; Schneider, R.; Schuster, R.

doi:10.1090/S1061-0022-07-00990-9

The space of isometry covariant tensor valuations
HTML articles powered by AMS MathViewer

by D. Hug, R. Schneider and R. Schuster

St. Petersburg Math. J. 19 (2008), 137-158

DOI: https://doi.org/10.1090/S1061-0022-07-00990-9

Published electronically: December 17, 2007

PDF | Request permission

Abstract:

It is known that the basic tensor valuations which, by a result of S. Alesker, span the vector space of tensor-valued, continuous, isometry covariant valuations on convex bodies, are not linearly independent. P. McMullen has discovered linear dependences between these basic valuations and has implicitly raised the question as to whether these are essentially the only ones. The present paper provides a positive answer to this question. The dimension of the vector space of continuous, isometry covariant tensor valuations, of a fixed rank and of a given degree of homogeneity, is explicitly determined. The approach is constructive and permits one to provide a specific basis.

References

S. Alesker, Continuous valuations on convex sets, Geom. Funct. Anal. 8 (1998), no. 2, 402–409. MR 1616167, DOI 10.1007/s000390050061
S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. (2) 149 (1999), no. 3, 977–1005. MR 1709308, DOI 10.2307/121078
S. Alesker, Description of continuous isometry covariant valuations on convex sets, Geom. Dedicata 74 (1999), no. 3, 241–248. MR 1669363, DOI 10.1023/A:1005035232264
S. Alesker, On P. McMullen’s conjecture on translation invariant valuations, Adv. Math. 155 (2000), no. 2, 239–263. MR 1794712, DOI 10.1006/aima.2000.1918
S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272. MR 1837364, DOI 10.1007/PL00001675
Semyon Alesker, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (2003), no. 1, 63–95. MR 2015260
S. Alesker, $\textrm {SU}(2)$-invariant valuations, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 21–29. MR 2087147, DOI 10.1007/978-3-540-44489-3_{3}
C. Beisbart, M. S. Barbosa, H. Wagner, and L. da F. Costa, Extended morphometric analysis of neuronal cells with Minkowski valuations, arXiv:cond-mat/0507648.
C. Beisbart, R. Dahlke, K. Mecke, and H. Wagner, Vector- and tensor-valued descriptors for spatial patterns, Morphology of Condensed Matter, Lecture Notes in Phys., vol. 600, Springer, Berlin, 2002, pp. 238–260.
H. Hadwiger and R. Schneider, Vektorielle Integralgeometrie, Elem. Math. 26 (1971), 49–57 (German). MR 283737
Daniel A. Klain, A short proof of Hadwiger’s characterization theorem, Mathematika 42 (1995), no. 2, 329–339. MR 1376731, DOI 10.1112/S0025579300014625
P. McMullen, Continuous translation-invariant valuations on the space of compact convex sets, Arch. Math. (Basel) 34 (1980), no. 4, 377–384. MR 593954, DOI 10.1007/BF01224974
Peter McMullen, Valuations and dissections, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933–988. MR 1243000
Peter McMullen, Isometry covariant valuations on convex bodies, Rend. Circ. Mat. Palermo (2) Suppl. 50 (1997), 259–271. II International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures” (Agrigento, 1996). MR 1602986
Peter McMullen and Rolf Schneider, Valuations on convex bodies, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 170–247. MR 731112
Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
A. V. Pukhlikov and A. G. Khovanskiĭ, Finitely additive measures of virtual polyhedra, Algebra i Analiz 4 (1992), no. 2, 161–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 337–356. MR 1182399
Rolf Schneider, On Steiner points of convex bodies, Israel J. Math. 9 (1971), 241–249. MR 278187, DOI 10.1007/BF02771589
Rolf Schneider, Krümmungsschwerpunkte konvexer Körper. I, Abh. Math. Sem. Univ. Hamburg 37 (1972), 112–132 (German). MR 307039, DOI 10.1007/BF02993906
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Rolf Schneider, Tensor valuations on convex bodies and integral geometry, Rend. Circ. Mat. Palermo (2) Suppl. 65 (2000), 295–316. III International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures”, Part I (Mazara del Vallo, 1999). MR 1809137
Rolf Schneider and Ralph Schuster, Tensor valuations on convex bodies and integral geometry. II, Rend. Circ. Mat. Palermo (2) Suppl. 70 (2002), 295–314. IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures $\&$ Applications to Engineering Science”, Vol. II (Tropea, 2001). MR 1962603