fom
Posts:
1,969
Registered:
12/4/12


Re: the range of values for "Pi".
Posted:
Feb 4, 2014 3:24 AM


On 2/3/2014 8:37 AM, quasi wrote: > quasi wrote: >> Niels Diepeveen wrote: >>> quasi wrote [edited]: >>>> >>>> Conjecture: >>>> >>>> Fix a norm on R^2. >>>> >>>> Let S = {v in R^2 : v = 1} where v denotes the norm >>>> of v. >>>> >>>> Thus S is the standard unit circle in R^2 relative to the >>>> given norm. >>>> >>>> Define the circumference C of S to be the sup of the >>>> perimeters of all convex polygons which can be inscribed >>>> in S (where the side lengths are computed using the given >>>> norm). >>>> >>>> For the given norm, define Pi as C/2. >>>> >>>> Some trivial observations: >>>> >>>> (1) For all norms on R^2, Pi >= 2 >>>> >>>> (2) For an appropriate choice of norm on R^2, Pi can be >>>> made equal to any specified real number between 2 and 4 >>>> inclusive. >>> >>> The last one does not seem trivial to me. >> >> The claim was based on a miscalculation. >> >>> 4 is attained when S is any parallelogram, >> >> Yes. >> >>> but how do you get 2 ? >> >> My mistake  I thought the max norm yielded Pi = 2, >> but actually, it yields Pi = 4, the same as for any >> parallelogram. >> >>> I can get 3 with a regular hexagon, >> >> Nice. >> >>> but then I'm out of ideas. >>> >>>> Conjecture: For all norms on R^2, Pi <= 4. >>> >>> Seems plausible, but I don't know how to prove it. >> >> Could be a rough one. >> >> So what we now know is this: >> >> (1) For all norms on R^2, Pi >= 2 >> >> (2) For an appropriate choice of norm on R^2, Pi can be >> made equal to any specified real number between 3 and 4 >> inclusive. >> >> Which suggests the following ... >> >> Conjecture: For all norms on R^2, 3 <= Pi <= 4. > > I can prove the following partial result ... > > For n even with n > 2, let Pi_n be the value of Pi when > S is a regular ngon centered at the origin. > > For n = 2 mod 4, the sequence Pi_n is strictly increasing > and converges to ordinary Pi. > > For n = 0 mod 4, the sequence Pi_n is strictly decreasing > and converges to ordinary Pi. > > It follows that > > min {Pi_n : n = 4,6,8, ...} = Pi_6 = 3 > > max {Pi_n : n = 4,6,8, ...} = Pi_4 = 4 > > quasi >
I can try to give you an account of your limits. I would hope someone has an easier explanation. But, here goes...
First there is a sequence of definitions that go round in circles to get back to the codomain of your norm  that is, R.
In "Functional Analysis" by Kantorovich the discussion of a norm traces back to the relationship between absorbent convex sets and gauge functions. This relationship describes a family of compatible seminorms based on convex sets.
Convexity is defined using the vector space relation,
z = kx + (1k)y
The limits are related to the '1k' term.
A set E is convex if z is in E whenever x and y are in E.
E is balanced if kx is in E whenever x is in E and k <= 1.
E is absolutely convex if kx + jy is in E whenever k <= 1, j <= 1, and both x and y are in E.
E is absorbent if for any vector v there is a k > 0 such that v is in jE for j with j >= k. So, an absorbent set E is a set having an interval on every ray passing through the origin.
Absorbent sets are associated with a single seminorm by a a greatest lower bound over the multipliers for the family of rays.
But, that is jumping a few definitions. To continue, a seminorm is a homogeneous gauge function. A gauge function is a semiadditive positive homogeneous function from the vector space into R.
Semiadditivity is defined as
p( x + y ) <= p(x) + p(y)
Positive homogeneity is defined as
p( k * x ) = k * p(x)
while homogeneity is defined as
p( k * x ) = k * p(x)
To every convex absorbent set E there corresponds the Minkowski functional,
p_E(x) = inf{ k : k > 0 and x in kE }
so that
{ x : p_E(x) < 1 } subset E subset { x : p_E(x) <= 1 }
p_E(x) is a seminorm if E is absolutely convex.
So, the usual definition of a norm,
 x  = 0 if and only if x = 0
 k * x  = k *  x 
 x + y  <=  x  +  y 
induces a topology on the vector space in terms of the family of seminorms associated with Minkowski functionals.
Now, "attaching" such a function is discussed in "Vectors and Tensors" by Bowen and Wang.
As had been pointed out to me in the past, there is a distinction between a mere vector space and an inner product space. If this construction is done with a vector space, it yields an affine space. If it is done with an inner product space, it yields a Euclidean point space.
The construction above requires a mapping into R. So, there are actually two constructions occurring here. First, R must be understood as an inner product space "attaching" to itself. Then, the domain for the gauge functions associated with a norm must be thought of as the set to which the inner product space R is being "attached".
Here is their definition:
<begin quote>
Consider an inner product space V and a set E. The set E is a Euclidean point space if there exists a function
f: E x E :=> V
such that:
1) f(x,y) = (f,z) + f(z,y) for all x,y,z in E
2) For every x in E and v in V there exists a unique element y in E such that
f(x,y) = v
<end quote>
They then write:
"A Euclidean point space is not a vector space but a vector space with inner product is made a Euclidean point space by defining
f(v_1, v_2) = v_1  v_2
for all v in V. For an arbitrary point space the function f is called the point difference, and it is customary to use the suggestive notation,
f(x,y) = x  y"
They then go on to define a distance function in terms of the inner product of their definition,
d(x,y) =  x  y  = [( x  y )( x  y )]^1/2
Going back to Kantorivich and the notions from functional analysis, attaching a norm to a topological vector space induces the norm topology. The norm topology is a locally convex space by virtue of its system of neighborhoods about the origin. And, a normed space is metrizable. Hence, the last step from Bowen and Wang is precisely the step that attaches a metric to a normed space.
Now, consider the formula with which this began,
z = kx + (1k)y
The definition of a Minkowski functional depended upon an absorbent set. This is a notion of betweenness. So, in the equation above x, y, and z must be distinct. Therefore k cannot be 0 or 1.
Consider the form,
y = ( z  kx ) / ( 1  k )
The primary concern here is what occurs in R since this is the codomain of a norm. And, since x and y are supposed to be arbitrary points in a set with respect to the construction of a Euclidean point domain, take x and y to be different from 0. Then, we may write,
y =  z  kx  / ( 1  k )
so that
y >  z  kx 
y >  z  kx  + ( k *  z  kx  )
y >  z  kx  + ( k *  z  kx  ) + ( k^2 *  z  kx  )
and so on
by virtue of
Sum[n=0..oo] x^n = 1/(1x)
The purpose for all of this has been to convince you that the limits from your analysis of the question arise from an infintary process. If I have made a mistake and wasted your time, then you have no need to continue. My experience, however, is that tracing these definitions through different subjects often leads to this kind of tedious analysis to identify the point at which "foundational" concerns must be brought to bear.
With that, consider the initial set of intervals from my recent post,
0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3
You will see both 1/3 and 1/4 in the sequence of intervals. Their multiplicative inverses are your limits.
Now, for me 1/3 had been significant in relation to Cantor sets. But, when you recently informed me about Wilson's theorem, I have been learning about primes as quickly as possible.
In this instance, 1/3 also relates to the MarkovLagrange spectrum. This spectrum is normally thought to converge at 3. However, there is a particularly relevant analysis in the paper,
http://arxiv.org/pdf/0906.0611v1.pdf
Roy formulates an analysis of the Markov numbers in such a way that he replaces the usual triples with 2 x 2 matrices. Working in the interval [0,1/3] he demonstrates that the Lagrange spectrum converges simultaneously with two conjugate values. What is important is that there is a simultaneous convergence of both linear and quadratic forms in this convergence.
The simultaneous convergence is important because area is nonlinear. This is the source of incommensurability. So, the fact that infinitary processes arise in your question follows from the problem of definition for irrational numbers.
Now, the value of 1/4 enters the sequence above from my personal considerations. The purpose of the DedekindCantor program had been to provide an exact identity criterion for real numbers. I went back and read Bolzano's proof of the intermediate value theorem. It is prior to modern methods, and, it depends upon a fixedpoint argument. So, the intervals above are based on a system of fixedpoints overlapping one another.
First, the Cantor intervals [0,1/3^i] are divided to have midpoints,
[ 0, ..., 1/( 2 * 3^i ), ..., 1/3^i ]
Next, the rightmost subinterval is divided to have a "between point" which is not necessarily related by a direct mathematical equation. So, the largest dyadic power that lies in the interval is used,
[ 1/( 2 * 3^i ), ..., 1/2^j, ..., 3^i ]
The leftmost subinterval is divided into 6 subintervals. My reasoning for this arose from reading Sierpinski's monograph on "Pythagorean Triangles". In the last chapter he discusses certain relationships associated with using Pythagorean triples to form parallelepipeds. It is significant that the numbers 5 and 2 (in all powers for 2) occur as algebraic limits associated with those constructions. So, I used that idea to partition the subinterval.
My intuitions on the matter seem to have been correct. I found this while preparing this reply:
http://en.wikipedia.org/wiki/Fivepoint_stencil
Although that construction appears to use an "initial" partition, it is actually a "final" configuration. If you look at what is actually being done, a neighborhood system is being generated around the least prime, namely 2.
Consider the "residual" partition in the first 3 iterations,
interval_1 :=> { ..., 33 } interval_2 :=> { 31, ..., 17 } [5 primes] [length: 14] [gap: 2] interval_3 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4] interval_4 :=> {} [0 primes] [length: null] [gap: null] interval_5 :=> { 7 } [1 prime] [length: 0] [gap: 4] interval_6 :=> {} [0 primes] [length: null] [gap: null] interval_7 :=> { 5 } [1 prime] [length: 0] [gap: 2] interval_8 :=> {} [0 primes] [length: null] [gap: null]
residual :=> { 2 } [1 prime] [length: 0] [gap: 3] omitted :=> { 3 } [1 prime] [length: 0] [gap: 0]
interval_1 :=> { ..., 131 } interval_2 :=> { 127, ..., 67 } [13 primes] [length: 60] [gap: 4] interval_3 :=> { 61, ..., 43 } [5 primes] [length: 18] [gap: 6] interval_4 :=> { 41, 37 } [2 primes] [length: 4] [gap: 2] interval_5 :=> { 29 } [1 prime] [length: 0] [gap: 8] interval_6 :=> { 23, 19 } [2 primes] [length: 4] [gap: 6] interval_7 :=> { 17 } [1 prime] [length: 0] [gap: 2] interval_8 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4]
residual :=> { 7, 5, 3, 2 } [4 primes] [length: 5] [gap: 4] omitted :=> { 31 } [1 prime] [length: 0] [gap: 0]
interval_1 :=> { ..., 521 } interval_2 :=> { 509, ..., 257 } [49 primes] [length: 252] [gap: 2] interval_3 :=> { 251, ..., 173 } [15 primes] [length: 78] [gap: 6] interval_4 :=> { 167, ..., 131 } [8 primes] [length: 36] [gap: 6] interval_5 :=> { 127, ..., 103 } [5 primes] [length: 24] [gap: 4] interval_6 :=> { 101, ..., 59 } [10 primes] [length: 42] [gap: 2] interval_7 :=> { 53, ..., 37 } [5 primes] [length: 16] [gap: 6] interval_8 :=> { 31, 29 } [2 primes] [length: 2] [gap: 6]
residual :=> { 23, ..., 2 } [9 primes] [length: 21] [gap: 6] omitted :=> {} [0 primes] [length: null] [gap: null]
It generates the system of neighborhoods,
{ { 2 }, { 7, 5, 3, 2 }. { 23, ..., 2 } }
Obviously, if the generation of partitions is taken to infinity, this is a particular system of neighborhoods converging to 2.
So, in fact, the '1/4' in the system of intervals,
0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3
is a terminal value just as '1/3' is a terminal value in Roy's paper.
Now, my other post simply dealt with generating primes. But, as always, I am working at the sense of a foundation which drives my researches. The system of intervals,
0 < 1/32 < 2/32 < 3/32 < 4/32 < 5/32 < 1/6 < 1/4 < 1/3
Should actually be viewed as halfopen intervals,
( 0, 1/32 ], ( 1/32, 2/32 ], ( 2/32, 3/32 ], ( 3/32, 4/32 ], ...
... ( 4/32, 5/32 ], ( 5/32, 1/6 ], ( 1/6, 1/4 ], ( 1/4, 1/3 ]
And, after the first partition, they are interlaced with oppositely directed halfopen intervals. So, the second partition,
0 < 1/128 < 2/128 < 3/128 < 4/128 < 5/128 < 1/18 < 1/16 < 1/9
is overlayed with the intervals,
[ 1/49, 6/49 ), [ 5/49, 24/49 )
where the denominator is the square of the largest prime from the corresponding residual. In this case, one has 7^2 as the denominators because the residual is
{ 7, 5, 3, 2 }
The sequence < 24/49, 264/529, 3120/6241, ... > approaches 1/2 while the the sequence of Cantor intervals < [ 0, 1/3 ], [ 0, 1/9 ], [ 0, 1/27 ], ... > is driving the sequence of midpoints to 0. That is,
< 24/49, 264/529, 3120/6241, ... > > 1/2
< 1/6, 1/18, 1/54, ... > > 0
Once again, the notion of "between points" is implmented. So, the second partition would actually look like,
0 < 1/128 < 2/128 < 3/128 < 4/128 < 1/49 < 5/128 ...
... < 1/18 < 1/16 < 5/49 < 1/9 < 6/49 < 24/49
Relative to the first partition, 1/4 is omitted since it demarcates an interval.
And, in each partition the endpoints for the squared residuals are supposed to be as near as possible to specific points. So, one has
5/49 < 1/9 < 6/49
4/128 < 1/49 < 5/128
with the first interval being as small as possible around the value 1/3^i and the second interval having the squared residual value as near to the larger endpoint as possible.
There is also this subinterval,
5/49 < 6/49 < 24/49
In the limit, then, one obtains
( 0, 1/2 )
But, in fact, the values 1/3 and 1/4 arise from the "infinite descent" intended to form the neighborhood system around 2.
Now, I could talk about the relationships to various Diophantine equations and ideas from Sierpinski's monograph on Pythagorean triangles that motivated this. But, I have written enough.
I know it is a lot. I should add that for every odd prime p, 4p is representable as the sum of four odd integral squares. That is where your modulo 4 arithmetic arises. With that in mind, you should look at Marion's theorem and Morgan's theorem,
http://mathworld.wolfram.com/MarionsTheorem.html
Also, with a handful of removable discontinuities, one gets the postive reals as the codomains of
r(x) = x , x in ( 0, 1/2 )
s(x) = 1/2 + ( 1/2  x ), x in ( 0, 1/2 )
t(x) = 1/( 1/2 + ( 1/2  x ) ), x in ( 0, 1/2 )
u(x) = 1/x , x in ( 0, 1/2 )
But, there are other ways of trying to extend it. And, I have to learn the algebra to put this all together properly.

