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It is shown that 2-dimensional subdivisions can be made regular by moving their vertices within parallel 1-dimensional spaces. As a consequence, any 2-dimensional subdivision is projected from the boundary complex of a 4-polytope. ...
This paper features two main contributions. On the one hand, it gives an impressive survey on the progress on the diameter problem, including the breakthrough of the author with his disproof of the Hirsch conjecture among many other recent results. On the ...
Abstract: A simplicial complex C on a d-dimensional configuration of n points is k-regular if its faces are projected from the boundary complex of a polytope with dimension at most d+k. Since C is obviously (n-d-1)-regular, the set of all integers k for wh ...
The great Swiss mathematician Ludwig Schläfli (1814-1895) left after his death more than three hundred and fifty notebooks. They include mathematical studies and new results, as well as works about classical mathematical texts and a priori more surprising ...
Imagine that you are blindfolded inside an unknown room. You snap your fingers and listen to the room’s response. Can you hear the shape of the room? Some people can do it naturally, but can we design computer algorithms that hear rooms? We show how to com ...
To every d-dimensional polytope P with centrally symmetric facets one can assign a “subway map” such that every line of this “subway” contains exactly the facets parallel to one of the ridges of P. The belt diameter of P is the maximum number of subway lin ...
The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is sa ...
We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point- o ...
We consider the recovery of an airline schedule after an unpredicted event, commonly called disruption, that makes the planned schedule unfeasible. In particular we consider the aircraft recovery problem for a heterogeneous fleet of aircrafts, made of regu ...
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 p ...
