Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 1

Convex Hulls of Spatial Polygons with a Fixed Convex Projection

Boris V. Dekster

Let $F$ be a convex n-gon in a horizontal plane of the Euclidean 3-space.
Consider its spatial variation under which its vertices move vertically
and let $F_*$ be the convex hull of such a variation. In the general
position, the boundary of $F_*$
splits naturally into the "bottom" $F^{\prime}$ and
the "top" $F^{\prime\prime}$.
The polyhedron $F^{\prime}$
($F^{\prime\prime}$)
has triangular faces and no
vertices inside. The projection of these faces on $F$ yields a
triangulation
$T^{\prime}$
($T^{\prime\prime}$)
of $F$. Obviously
$T^{\prime}$ and
($T^{\prime\prime}$)
have no common diagonals.
Suppose now that
$T^{\prime}$ and
($T^{\prime\prime}$)
with no common diagonals are prescribed. 
Guibas conjectured that the appropriate variation exists. The present
paper gives a sufficient and a necessary condition of such existence.
The necessary condition can fail which disproves the Guibas conjecture.