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removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info; if there are some other geometry-related tags which are suitable, please use some of them instead
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edit approved Jun 14, 2017 at 5:27
Martin Sleziak
edit approved Jun 14, 2017 at 5:27
Martin Sleziak
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Characterizing Convex Configurations of QuadrupelsQuadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Improved formulation of problem statement and description in following a suggestion of Günter Rote
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Manfred Weis
Manfred Weis
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Characterizing Convex QuadrilateralsConfigurations of Quadrupels of Coplanar Points via Length Measurements and Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the 6 lengths ofeuclidean plane, is it possible to determine from the sides6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and diagonals of, $BD$ alone, whether every point is a planar quadrilateral pluscorner of their adjacency relations. Is it possible,convex hull under the restriction that comparing only sums orand/or differences of those lengths, to determine whether the quadrilateraldistances is convex?the only allowed operation and that no information about the point's coordinates is available.

Background of the question is whether it is possible to generalize the concept of intersection to non-adjacent edges of a general weighted graph. If the answer were affirmative, then geometric concepts like inside/outside relations or convex hulls would be an "intrinsic" property of graphs; triangulations of complete graphs would then also be more flexible and could contain complete subgraphs of order 4.

Characterizing Convex Quadrilaterals via Length Measurements and Linear (In-)equalities

Given the 6 lengths of the sides and diagonals of a planar quadrilateral plus their adjacency relations. Is it possible, comparing only sums or differences of those lengths, to determine whether the quadrilateral is convex?

Background of the question is whether it is possible to generalize the concept of intersection to non-adjacent edges of a general weighted graph. If the answer were affirmative, then geometric concepts like inside/outside relations or convex hulls would be an "intrinsic" property of graphs; triangulations of complete graphs would then also be more flexible and could contain complete subgraphs of order 4.

Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a corner of their convex hull under the restriction that comparing sums and/or differences of the distances is the only allowed operation and that no information about the point's coordinates is available.

Background of the question is whether it is possible to generalize the concept of intersection to non-adjacent edges of a general weighted graph. If the answer were affirmative, then geometric concepts like inside/outside relations or convex hulls would be an "intrinsic" property of graphs; triangulations of complete graphs would then also be more flexible and could contain complete subgraphs of order 4.

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Manfred Weis
Manfred Weis
  • 13.2k
  • 4
  • 34
  • 76

Characterizing Convex Quadrilaterals via Length Measurements and Linear (In-)equalities

Given the 6 lengths of the sides and diagonals of a planar quadrilateral plus their adjacency relations. Is it possible, comparing only sums or differences of those lengths, to determine whether the quadrilateral is convex?

Background of the question is whether it is possible to generalize the concept of intersection to non-adjacent edges of a general weighted graph. If the answer were affirmative, then geometric concepts like inside/outside relations or convex hulls would be an "intrinsic" property of graphs; triangulations of complete graphs would then also be more flexible and could contain complete subgraphs of order 4.