Convex hull of the maximum volume of a space curve in the special case
Vladimir Shelomovskii
Deoma, Russian Federation
Let СN be the closed three-dimensional polygon with 2N edges (N > 3), the perimeter L(СN) and the convex hull of СN volume V(CN). We want to find maximum V(CN) for given L(СN) in the special case when the convex hull may be divided into tetrahedra having one common edge. Let C be the rectifiable closed three-dimensional curve with the length L(C) and the volume of the convex hull V(C). C may be obtained using the СN limit at infinity. We want to find maximum V(C) for given L(C).
We assume that the convex hull maximum volume is achieved if two conditions are satisfied: at first, the slope angle θ between the curve and the Z-axis is constant, the segment which intersects the Z-axis is perpendicular to it. The second condition is: the projection of the convex hull to the XY plane has the maximum area. For V(L) there is an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper. There are two solids of equal volume. One solid is axisymmetric, the second solid is centrally symmetric.
The area of the convex hull projection on the XY plane has an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper.
Finding the maximum area of the convex hull of CN projection in case N = 4 + 2n is reduced to finding the function extremum under the obtained conditions. These equations may be solved analytically for n < 12 and numerically for an arbitrary n. The solution has been found and checked using DGS GInMA. There are some examples of the maximum volume convex hulls V(CN) and maximum V(C) for given L(C).