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CATEGORIES:Optimisation and Numerical Analysis Seminars
SUMMARY:Volumes of alcoved polyhedra and Mahler's conjectu
re - Maria Jesus de la Puente (Madrid\, Spain)
DTSTART:20190306T120000Z
DTEND:20190306T130000Z
UID:TALK3548AT
URL:/talk/index/3548
DESCRIPTION:There is a conjecture\, by K. Mahler\, open since
1939 in dimensions 3 and higher: every centrally
symmetric convex body K satisfies the inequality
"vol(K) vol(K') >= 4^n/n!" where K' is the po
lar of K. The conjecture means that the volum
e product vol(K) vol(K') is minimal when K is
an n-dimensional cube (or an affine transform o
f it). On the other hand\, the Blaschke֭-Santal
o inequality (proved in 1949) says that the volu
me product is maximal when K is\nan n-dimensiona
l sphere (or an ellipsoid\, an affine transform
).\n\n\nA polytope is the generalization to n-
dimensional space of a polygon (n=2) and a polyh
edron (n=3). Alcoved polytopes are polytopes hav
ing facets of only two types: x_i=cnt and x_i-x_
j=cnt. Each n-dimensional alcoved polytope P ca
n be represented by a square matrix A(P) of or
der n+1. The matrix A(P) can be assumed to be m
ultiplicatively idempotent with respect to tropic
al product\, where "max" plays the role of addit
ion (tropical sum) and "plus" plays the role of mu
ltiplication (tropical product). The volume vol(P
) is obtained as an expression in terms of the en
tries A(P). In the case n=3\, we have found a gen
eral formula for such expression. Further\, for ce
ntrally symmetric P\, we have found another formul
a for the volume product vol(P)vol(P') (after an
affine change). This is a rational expression in
three variables (with rational coefficients). Mahl
er's Conjeture in this case reduces to the statem
ent that a certain polynomial\, which is non-homog
eneous\, of degree 6\, in three variables\, is non
--negative over a given tetrahedron. We will show
that this holds true.\n\nIn October 2017\, a sec
ond version of a proof of Mahler Conjecture has b
een uploaded to Arxiv\, for n=3\, by Iriyeh and
Shibata (a first version in June 2017).\n
LOCATION:Room R17/18\, Watson building
CONTACT:Sergey Sergeev
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