Re: Generalized convex perimeter of an n-d object

On Mar 30, 11:25 am, I wrote:

If S is a sphere, then p_3(S) is 4pi times its radius. If S is a
polyhedron, then p_3(S) is a weighted sum of the lengths of its edges,
where the weight of each edge is the supplement of the dihedral angle
at that edge (i.e., pi - the angle). Thus, one could pose the
problem, "prove that this funny weighted sum is between 4pi*r and
4pi*R where r and R are the radii of the inscribed and circumscribed
sphere, respectively."

Correction: the edge weight is half of what I wrote, i.e., (pi - the
dihedral angle)/2

Is there a nice integral expression for the p_n(S) when S is convex
and smooth?

After looking at some special cases, I'm fairly confident in the
following conjecture (no proof though):

For any bounded S in R^n with smooth convex hull c(S)

p_n(S) = integral over the boundary of c(S) K_(n-2) dA,

where dA is an element of (n-1)-d surface area, and (the invariant
curvature) K_j is defined (cf. http://www.mathpages.com/home/kmath520/kmath520.htm)
as the average of all products of j distinct principal curvatures.
I.e.,

p_2(S) = intg dA = perimeter of c(S)
p_3(S) = intg (k_1 + k_2)/2 dA
p_4(S) = intg (k_1 k_2 + k_1 k_3 + k_2 k_3)/3 dA
p_5(S) = intg (k_1 k_2 k_3 + k_1 k_2 k_4 + k_1 k_3 k_4 + k_2 k_3 k_4)/
4 dA
etc.

So this would make a nice problem to spring on someone: using the
above conjecture to define p_n(S) (rather than the definition in the
previous post), show that if A and B are bounded, convex, smooth
subsets of R^n, with A contained in B, then p_n(A) <= p_n(B).

-Jim Ferry
Metron, Inc.
f rr @m tsc .c m
e y e i o

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