14 research outputs found
Topological obstructions for vertex numbers of Minkowski sums
We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i
\ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the
maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of
Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands.
The result is obtained by combining methods from discrete geometry (Gale
transforms) and topological combinatorics (van Kampen--type obstructions) as
developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos.
Construction/example and remarks adde
The maximum number of faces of the Minkowski sum of three convex polytopes
We derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves
Non-projectability of polytope skeleta
We investigate necessary conditions for the existence of projections of
polytopes that preserve full k-skeleta. More precisely, given the combinatorics
of a polytope and the dimension e of the target space, what are obstructions to
the existence of a geometric realization of a polytope with the given
combinatorial type such that a linear projection to e-space strictly preserves
the k-skeleton. Building on the work of Sanyal (2009), we develop a general
framework to calculate obstructions to the existence of such realizations using
topological combinatorics. Our obstructions take the form of graph colorings
and linear integer programs. We focus on polytopes of product type and
calculate the obstructions for products of polygons, products of simplices, and
wedge products of polytopes. Our results show the limitations of constructions
for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge
product surfaces of R\"orig & Ziegler (2009) and complement their results.Comment: 18 pages, 2 figure
Construction and Analysis of Projected Deformed Products
We introduce a deformed product construction for simple polytopes in terms of
lower-triangular block matrix representations. We further show how Gale duality
can be employed for the construction and for the analysis of deformed products
such that specified faces (e.g. all the k-faces) are ``strictly preserved''
under projection. Thus, starting from an arbitrary neighborly simplicial
(d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose
projection to the last dcoordinates yields a neighborly cubical d-polytope. As
an extension of thecubical case, we construct matrix representations of
deformed products of(even) polygons (DPPs), which have a projection to d-space
that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both
cases the combinatorial structure of the images under projection is completely
determined by the neighborly polytope Q: Our analysis provides explicit
combinatorial descriptions. This yields a multitude of combinatorially
different neighborly cubical polytopes and DPPs. As a special case, we obtain
simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler
(2000) as well as of the ``projected deformed products of polygons'' that were
announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets
arbitrarily close to 9.Comment: 20 pages, 5 figure
Prodsimplicial-Neighborly Polytopes
Simultaneously generalizing both neighborly and neighborly cubical polytopes,
we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to
that of a product of r simplices. We construct PSN polytopes by three different
methods, the most versatile of which is an extension of Sanyal and Ziegler's
"projecting deformed products" construction to products of arbitrary simple
polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1.
Using topological obstructions similar to those introduced by Sanyal to bound
the number of vertices of Minkowski sums, we show that this dimension is
minimal if we additionally require that the PSN polytope is obtained as a
projection of a polytope that is combinatorially equivalent to the product of r
simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction
The maximum number of faces of the Minkowski sum of two convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of two
-dimensional convex polytopes and , as a function of the number
of vertices of the polytopes.
For even dimensions , the maximum values are attained when and
are cyclic -polytopes with disjoint vertex sets. For odd dimensions
, the maximum values are attained when and are
-neighborly -polytopes, whose vertex sets are
chosen appropriately from two distinct -dimensional moment-like curves.Comment: 37 pages, 8 figures, conference version to appear at SODA 2012; v2:
fixed typos, made stylistic changes, added figure
A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of convex
-polytopes in , where and , as a
(recursively defined) function on the number of vertices of the polytopes.
Our results coincide with those recently proved by Adiprasito and Sanyal [2].
In contrast to Adiprasito and Sanyal's approach, which uses tools from
Combinatorial Commutative Algebra, our approach is purely geometric and uses
basic notions such as - and -vector calculus and shellings, and
generalizes the methodology used in [15] and [14] for proving upper bounds on
the -vector of the Minkowski sum of two and three convex polytopes,
respectively.
The key idea behind our approach is to express the Minkowski sum
as a section of the Cayley polytope of the
summands; bounding the -faces of reduces to bounding the
subset of the -faces of that contain vertices from each
of the polytopes.
We end our paper with a sketch of an explicit construction that establishes
the tightness of the upper bounds.Comment: 43 pages; minor changes (mostly typos