In finite dimensional Finsler geometry, we define Minkowski functional on tangent spaces that are finite dimensional vector spaces. The definition of Minkowski functional can be generalized to infinite dimensional vector spaces that admit inner product at least but I have not found in literature such a generalization. Is there a problem with using this definition in infinite dimensional spaces?.
Asked
Viewed
82 times

$\begingroup$ I imagine the problem is the definition of volume (and its derivatives) for linear combinations of convex bodies in infinite dimensions. Deane Yang or Semyon Alesker will know more about this. $\endgroup$ – alvarezpaiva Jul 23 '19 at 7:53

$\begingroup$ Thanks, I am interested not in full generality of finsler geometry, mainly for defining the length of curves and perhaps continuity of norms suffices . $\endgroup$ – Richard Kim Jul 23 '19 at 9:56

$\begingroup$ Unfortunately I don’t know anything about this. I’ve never thought about the infinite dimensional case, where the challenges and questions are usually quite different from the finite dimensional case. Maybe some of our friends in Banach space theory would know something about this? $\endgroup$ – Deane Yang Jul 23 '19 at 13:30

$\begingroup$ Sorry, I think i mixed up "mixed volumes" with "Minkowski functional" (i.e. the possibly asymmetric norm associated to a convex body containing the origin in its interior). What is it exactly you want? Do you have closed bounded convex sets which contain a nbd of the origin in each tangent space and want to get a Finsler metric from that? $\endgroup$ – alvarezpaiva Jul 23 '19 at 16:03

$\begingroup$ @alvarezpaiva, I'm always confused by what the "Minkowski functional" is. $\endgroup$ – Deane Yang Jul 23 '19 at 16:08