The space of tropically collinear points is shellable

The Space of Tropically Collinear Points Is Shellable

The space Td,n of n tropically collinear points in a fixed tropical projective space TP is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d × n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space M0,n(TP , 1) of n-marked tropical ...

The Complement of a D-Tree is Pure Shellable

Collinear Points in Permutations

Consider the following problem: how many collinear triples of points must a transversal of Zn×Zn have? This question is connected with venerable issues in discrete geometry. We show that the answer, for n prime, is between (n− 1)/4 and (n − 1)/2, and consider an analogous question for collinear quadruples. We conjecture that the upper bound is the truth and suggest several other interesting pro...

Many collinear k-tuples with no k+1 collinear points

For every k > 3, we give a construction of planar point sets with many collinear k-tuples and no collinear (k + 1)-tuples.

Space Debris Hazards from Explosions in the Collinear Sun-earth Lagrange Points

The collinear Lagrange points of the Sun-Earth system provide an ideal environment for highly sensitive space science missions. Consequently many new missions are planed by ESA and NASA that require satellites close to these points. For example, the SOHO spacecraft built by ESA is already installed in the first collinear Lagrange point. Neither uncontrolled spacecraft nor escape motors will sta...