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DESCRIPTION: The study of realization spaces of convex polytopes is one of
the oldest subjects in Polytope Theory. Most likely\, it goes back to Legen
dre (1794). A lot of progress took place since that time. However\, many q
uestions remained open. In general\, computing the dimension of the realiza
tion space $\mathcal{R}(P)$ of a d-polytope $P$ is hard\, even for $d = 4$\
, as shown by Mnëv (1988) and Richter-Gebert (1996). In this presentation\
, we will discuss two criteria to determine the dimension of the realizatio
n space\, and use them to show that $\dim \mathcal{R}(P) = f_1(P) + 6$ for
a 3-polytope $P$\, and $\dim \mathcal{R}(P) = df_{d-1}(P)$ (resp. $\dim \ma
thcal{R}(P) = df_0(P)$ ) for a simple (resp. simplicial) $d$-polytope $P$.
We will also discuss the realization spaces of some interesting 2-simple 2-
simplicial 4-polytopes. Namely\, we will consider the realization space of
the 24-cell and of a 2s2s polytope with 12 vertices which was found by Miya
ta (2011)\, and give a better bound for/determine its dimension.
DTSTAMP:20181105T173500
DTSTART:20180702T160000
CLASS:PUBLIC
LOCATION:Freie Universität Berlin \n Institut für Informatik \n Takustr. 9
\n 14195 Berlin \n Room 005 (ground floor)
SEQUENCE:0
SUMMARY:Laith Rastanawi (Freie Universität Berlin): On the Dimension of the
Realization Spaces of Polytopes
UID:89792711@www.facetsofcomplexity.de
URL:http://www.facetsofcomplexity.de/monday/20180702-C-Rastanawi.html
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