New conjecture on Corner Geometry
- From: We Pretty <we_pretty@xxxxxxxxx>
- Date: Thu, 7 Feb 2008 08:15:07 -0800 (PST)
More on corners can be found in:
http://groups.google.fi/group/sci.math/browse_frm/thread/761ea590d0110db5/4f68309e6fb3a5e8?hl=en&lnk=st&q=#4f68309e6fb3a5e8
shortcut:
http://tinyurl.com/yvxq6b
but I include the neccessary definitions for
the conjecture
Let M = (X, d) be a metric space.
Definition of corner point:
Let S be subset of X. p is a corner point of S in X iff
there exists a sequence B_1, B_2, B_3,... of open balls such that:
1. the elements in the sequence are pairwise disjoint
2. for each B_i:
2.1 there exists (at least) two distinct points such that
both are boundary points of the closure of S and
both are boundary points of the closure of B_i
2.2 the intersection of B_i and boundary of S is empty
2.3 the intersection of the closure of B_i and the closure of
B_i+1 is a singleton and not in the closure of S
3. each sequence p_1, p_2, p_3, ... where each p_i is in B_i, has
p as a limit point
Definition of corner set: C(S) = { c | c is a corner point of S }
Definition of corner sequence C_n of S:
C_0 = S
C_n+1 = C(C_n(S))
Definition of corner rank:
If there exists an n such that C_n(S) = the empty set,
then corner rank of S is smallest n such that C_n(s) = empty set,
otherwise S has infinite corner rank
Conjecture:
Let S be a set in R^n with the usual metric with finite corner
rank and having integer Hausdorff-dimension D. Then the
Hausdorff-dimension of C(S) is an integer.
I would very much appreciate refutation, proof or hints how to
prove or refute.
We Pretty
.
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