Dear Herve, I'am interested on your topic: do you have a paper/document on theorem's proof? Thanks.  Gianluca

Actually, for the moment no.

At this past time, near the year 1988 when I was student, I have used simply a pen ;-) for writting the report for the project client (because this was a contractual project from which I have won money : it was welcome for my student live ;-)

But if there are more interest about the subject I will certainly write a technical report, promise.

But before

1/ I need to find my lost report ;-) (as Ramanujan ;-), yes I do not find the report at home for the momet !!!!! But do not worry, I will recover it. If not, I will be able to rewrite the paper, there are only TWO KEY ideas in fact. The remaining is relatively simple.

2/ I must be sure that it is not an obvious consequence of convex maths ?

Thx for your interest

Cheers,

Hervé

Herve, I think that your result could be true, and that it is a consequence of using convex body. Do you have seen the Prof. Buttazzo (RG member) papers on Newton's problem of Minimal Resistence? The physical model is not necessary for your result, which is a geometric property, but I suspect that in his related work on bodies resistence, Newton already discovered this result. Herve, you are in good company!  Gianluca

Thx for this way, I gonna looking at this.

Let's keep in touch

Cheers

Hervé

Dear Gianluca,

Thank you for the idea, but I have read several papers about this Newton problem, and I really do not think there can be a relationship wuth my problem.

It is not the consequence of the great difference between the two formulation, but either the fact that the Newton problem :

1/ First study only revolution symmetry body

2/ Second, the projection is made only on the symmetry axis

I continue my research about the subject.

Now I know the real english term for what I called "master couple", it is called the cross sectionnal surface area.

Thank you for helping me,

I will inform you of the evolution of the situation.

Cheers,

Hervé

For all followers, just FYI

I think (with a high level of probability) that the technics used in my proof of the theorem could be generalized in higher dimensions.

Hervé POULARD

Dear Gianluca,

I inform you that I have recovered my "lost report" (with huge amount of dust above ;-)), and considering that I think that I have made sufficient research in the available litterature I have decided to write a paper.

But due to my professional activities and also other personnal interests, It may take time. So be patient !

Cheers,

Hervé

@ Herve Poulard: ... I have decided to write a paper.

Your idea is worthwhile and very interesting. You may want to consider introducing one or two Lemmas (helping theorems) as a runup to the proof of your theorem.

Perhaps, you will find the following paper on surface area (related to cross sectionnal surface area ) and convexity interesting:

Polar Duals of Convex Bodies,

Mostafa Ghandehari,Proceedings of the American Mathematical SocietyVol. 113, No. 3 (Nov., 1991), pp. 799-808Published by: American Mathematical SocietyDOI: 10.2307/2048618Stable URL: http://www.jstor.org/stable/2048618 (available online free)

This paper is a good one to check in terms of proofs related to your interests.

Here is a good paper on convex bodies with good examples of theorems and proofs:

Some Means of Convex Bodies

W. J. FireyTransactions of the American Mathematical SocietyVol. 129, No. 2 (Nov., 1967), pp. 181-217Published by: American Mathematical SocietyDOI: 10.2307/1994373Stable URL: http://www.jstor.org/stable/1994373Page Count: 37

Here is a good paper with proofs of simple theorems (called Propositions) in terms of surface area and convexity:

Acta Appl Math (2014) 129:81–134 DOI 10.1007/s10440-013-9831-6 A Plasticity Principle of Convex Quadrilaterals on a Convex Surface of Bounded Specific Curvature Anastasios N. Zachos

https://link-springer-com.uml.idm.oclc.org/content/pdf/10.1007%2Fs10440-013-9831-6.pdf

Here is a related paper with detailed theorems and proofs:

arXiv:1611.08921 [pdf, ps, other]

Estimating volume and surface area of a convex body via its projections or sections Alexander Koldobsky, Christos Saroglou, Artem Zvavitch

I look forward to reading your report.

Dear Professor,

Thank you very very much for the interest and the links

I will keep you informed about the subject.

Cheers,

Hervé

Hi,

I guess you are talking about the so-called Archimedes' Hat-Box Theorem,

http://mathworld.wolfram.com/ArchimedesHat-BoxTheorem.html

Fabrizio

As a byproduct you may think of the surface area of a sphere of radius R equal to the circumference of a cylinder of radius R, i.e. (2* R* pi), multiplied by height , i.e. (2 * R), which is 4 * pi * R**2 .

Fabrizio

Hello,

No I do not think that there is any relationship with this theorem, which particular case, and even do not concerned the problem : the value os the "mean projection surface area", called Ms, fonction of the exterior surface of the body, Surf

The case of the sphere if of course trivial as the sphere as always the same projection surface area : pi * R^2 and then the average is also pi * R^2, so while the surface of the sphere is 4 pi R^2, then we have Ms = Surf / 4 = pi * R^2

But the case of the cylinder is more complex : this is

Ms = pi * R^2 / 2 + pi R h / 2

it depends on the two parameters R and h, of course.

Best regards.

Hervé

Hello,

it might be interesting to take a look at this recent work

Article Inequalities for the surface area of projections of convex bodies

Fabrizio

Dear Professor,

Thanks so much for this reference. It is apparently (because I have not deeply studied all papers) fully in concern of my theorem.

Sincerely yours

Herve POULARD

Dear all followers,

Here is the correct (I hope ;-) ) generalization for n dimensions of my theorem.

Lemme 1 : The proof principle can be easily used in any dimension

(as to be written)

Lemme 2 : The way the theroem is proven imply that for any dimension there exist a coefficient a(n) for which the Ms (mean projection surface area of the convex body) can be written in the form of : Ms(n) = Surf(n) / a(n)

(as to be written)

Lemme 3 : For any dimension the hypersphere has always the same projection then the Ms is that number. (obvious)

Lemme 4 : For a n-euclidean dimension hypersphere the projection surface is the volume of the (n-1) dimension of a hypersphere, and the Surf(n) is the classical surface of the (n)-dimensional hypersphere

(obvious)

===========================================================

Theoreme :

The coefficient a(n) can be express with the following formula

a(2) = pi

for any n >= 3

a(n) = S(n) / V(n-1)

(where Sn is the surface of the n-dimpension hypersphere and V(n-1) the volume of the (n-1) hypersphere)

so concretly it can be expressed with the gamma function as :

a(n) = 2 . sqrt(PI) * gamma((n-1)/2) / gamma(n/2)

===========================================================

I have not elucided why the formula is noit valid for dim 2 ?

Otherwise we get the result :

a(2) = pi = 3.1415

a(3) = 4

a(4) = pi = 3.1415

a(5) = 8/3 = 2.666666666....

a(6) = 3/4*pi = 2.35....

....

The curve is maximale at dimension 3, decrease rapidely at the beginning and very slowly after, I have not yet analyse if it converges to zero or not ?

Thanks you for following this, and if I have made a mistake, please do not hesitate to correct me. I have made this rather rapidely !

Sincerely yours, and many thanks in advances for all your comments, corrections and addons !

Hervé POULARD

oups sorry for my poor english ;-)

Dear Followers,

The limit of the a(n) coefficient at infinity is 0.

Using the asympotic expression of the gamma function we get

a(x) =

(2 Sqrt[2 E \[Pi]] (-1 + x)^(-4 + x/2) x^(

7/2 - x/2) (-5629 + 17370 x - 18360 x^2 + 6480 x^3))/(-139 + 90 x +

1080 x^2 + 6480 x^3)

In this expression we can remove the polynomials terms as at infinity the ratio of the x^3 terms converge to a constant.

If we remove also the other constants the interesting terms is :

(-1 + x)^(-4 + x/2) x^(7/2 - x/2)

Here it is more tricky, let us take the ln

It gives also with some arrangements :

[ x/2*ln(x-1) - x/2*ln(x) ] + [7/2 ln(x) - 4 ln(x-1) ]

As ln(x-1) is equivalent at infinity to ln(x) then we have

3.5 ln(x) - 4 ln(x) = - 0.5 ln(x) which converges to - infinity

Therefore the a(n) converges to 0.

Is that correct ?

I have also verified graphically with Mathematica, and numerically for the value of a(n) with n from 2 to 6 (latest post). There is no doubt.

BUT I do no elucidate why the analytic expression of a(x) is no exact for dimension 2. There is this factor 2 before Pi which must not be here !

It may be an obvious reasonning error from me ????

The ln(a(x)) becomes negative in the interval [26.6,26.65]

Thanks in advances for any comments ....

Sincerely yours

Herve POULARD

Just a remark,

Concerning the approximation using the stirling formula for asympotic gamma : at x = 26, it is very accurate.

The ln(a(x)) ln(approximate function) becommes negative in the same interval mentionned last post [26.6,26.65]

Now, I think that the "game is done" and

I will try to write a first prepint of my paper as soon as possible.

Best Regards

Herve POULARD

good subject

Thank you Ola ;-)

Have a nice day.

Hervé