For convex sets in the Lorentzian Minkowski space bounded by space-like hyperplanes, it is possible to define area measures, similarly to the classical definition for convex bodies in the Euclidean space. Here the measures are defined on the hyperbolic space rather than on the round sphere. We are particularly interested by convex sets invariant under the action of isometries groups of the Minkowski space, so that the measures can be defined on compact hyperbolic manifolds. We can then look at the Christoffel and the Minkowski problems (i.e. particular measures are prescribed) in a general setting. In dimension , the Christoffel problem include a famous construction by G. Mess. In this dimension, the smooth version of the Minkowski problem already had a positive answer, and we show that this is a specificity of dimension , while the general problem has a solution in all dimensions.
@article{TSG_2014-2015__32__97_0, author = {Fillastre, Fran\c{c}ois}, title = {Christoffel and {Minkowski} problems in {Minkowski} space}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {97--114}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {32}, year = {2014-2015}, doi = {10.5802/tsg.305}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.305/} }
TY - JOUR AU - Fillastre, François TI - Christoffel and Minkowski problems in Minkowski space JO - Séminaire de théorie spectrale et géométrie PY - 2014-2015 SP - 97 EP - 114 VL - 32 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.305/ DO - 10.5802/tsg.305 LA - en ID - TSG_2014-2015__32__97_0 ER -
%0 Journal Article %A Fillastre, François %T Christoffel and Minkowski problems in Minkowski space %J Séminaire de théorie spectrale et géométrie %D 2014-2015 %P 97-114 %V 32 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.305/ %R 10.5802/tsg.305 %G en %F TSG_2014-2015__32__97_0
Fillastre, François. Christoffel and Minkowski problems in Minkowski space. Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 97-114. doi : 10.5802/tsg.305. http://www.numdam.org/articles/10.5802/tsg.305/
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