1,130 research outputs found
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
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
On f-vectors of Minkowski additions of convex polytopes
The objective of this paper is to present two types of results on Minkowski
sums of convex polytopes. The first is about a special class of polytopes we
call perfectly centered and the combinatorial properties of the Minkowski sum
with their own dual. In particular, we have a characterization of face lattice
of the sum in terms of the face lattice of a given perfectly centered polytope.
Exact face counting formulas are then obtained for perfectly centered simplices
and hypercubes. The second type of results concerns tight upper bounds for the
f-vectors of Minkowski sums of several polytopes.Comment: 13 pages, submitted to Discrete & Computational Geometr
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