Page 48 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 48
PLENARY SPEAKERS
Z(K) = c0V0(K) + · · · + cnVn(K)
for every K ∈ Kn.
Here V0(K), . . . , Vn(K) are the intrinsic volumes of K ∈ Kn. In particular, V0(K) is the
Euler characteristic of K, while 2 Vn−1(K) is the surface are of K and Vn(K) the n-dimensional
volume of K. Hadwiger’s theorem shows that the intrinsic volumes are the most basic function-
als in Euclidean geometry. It finds powerful applications in Integral Geometry and Geometric
Probability.
The fundamental results of Blaschke and Hadwiger have been the starting point of the de-
velopment of Geometric Valuation Theory. Classification results for valuations invariant (or co-
variant) with respect to important groups are central questions. The talk will give an overview
of such results including recent extensions to valuations on function spaces.
References
[1] K.J. Böröczky, M. Ludwig, Minkowski valuations on lattice polytopes. J. Eur. Math. Soc.
(JEMS) 21 (2019), 163–197.
[2] A. Colesanti, M. Ludwig, F. Mussnig, Minkowski valuations on convex functions, Calc.
Var. Partial Differential Equations 56 (2017), 56:162.
[3] A. Colesanti, M. Ludwig, and F. Mussnig, The Hadwiger theorem on convex functions. I,
arXiv:2009.03702 (2020).
[4] C. Haberl and L. Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27
(2014), 685–705.
[5] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin,
1957.
[6] D. A. Klain and G.-C. Rota, Introduction to geometric probability, Cambridge University
Press, Cambridge, 1997.
[7] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158–168.
[8] M. Ludwig, Valuations on Sobolev spaces, Amer. J. Math. 134 (2012), 827–842.
[9] M. Ludwig and M. Reitzner, A characterization of affine surface area, Adv. Math. 147
(1999), 138–172.
[10] M. Ludwig and M. Reitzner, A classification of SL(n) invariant valuations, Ann. of Math.
(2) 172 (2010), 1219–1267.
46
Z(K) = c0V0(K) + · · · + cnVn(K)
for every K ∈ Kn.
Here V0(K), . . . , Vn(K) are the intrinsic volumes of K ∈ Kn. In particular, V0(K) is the
Euler characteristic of K, while 2 Vn−1(K) is the surface are of K and Vn(K) the n-dimensional
volume of K. Hadwiger’s theorem shows that the intrinsic volumes are the most basic function-
als in Euclidean geometry. It finds powerful applications in Integral Geometry and Geometric
Probability.
The fundamental results of Blaschke and Hadwiger have been the starting point of the de-
velopment of Geometric Valuation Theory. Classification results for valuations invariant (or co-
variant) with respect to important groups are central questions. The talk will give an overview
of such results including recent extensions to valuations on function spaces.
References
[1] K.J. Böröczky, M. Ludwig, Minkowski valuations on lattice polytopes. J. Eur. Math. Soc.
(JEMS) 21 (2019), 163–197.
[2] A. Colesanti, M. Ludwig, F. Mussnig, Minkowski valuations on convex functions, Calc.
Var. Partial Differential Equations 56 (2017), 56:162.
[3] A. Colesanti, M. Ludwig, and F. Mussnig, The Hadwiger theorem on convex functions. I,
arXiv:2009.03702 (2020).
[4] C. Haberl and L. Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27
(2014), 685–705.
[5] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin,
1957.
[6] D. A. Klain and G.-C. Rota, Introduction to geometric probability, Cambridge University
Press, Cambridge, 1997.
[7] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158–168.
[8] M. Ludwig, Valuations on Sobolev spaces, Amer. J. Math. 134 (2012), 827–842.
[9] M. Ludwig and M. Reitzner, A characterization of affine surface area, Adv. Math. 147
(1999), 138–172.
[10] M. Ludwig and M. Reitzner, A classification of SL(n) invariant valuations, Ann. of Math.
(2) 172 (2010), 1219–1267.
46
