Hull Last

Is the convex hull about strawberries? I.M.R. Pinheiro∗ December 5, 2009

Abstract In this paper, we explain and fix the notions of both polynomially convex hull and hull. We also present the axioms of definition for both calyx and holder.

AMS (2000): 26A51 Key-words: Analysis, Convexity, Definition, hull, polynomially.

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Introduction

Polynomially convex hull seems to be a very interesting notion, especially when the words forming its referent are taken into consideration. Hull is a referent that would be usually associated with some fruits, or even seeds, such as the grain of rice, according to [6] or [5], for instance (in [5], we find strawberry hull being described as ‘persistent enlarged calyx at base’). Pushing it a bit, the support to the leaves of a flower or the contact surface of the stem with the leaf could be called hull. Because of the above facts, one would expect the external case of a set, as in a notebook case, to be present in a ‘hull’, or the top support to something, once strawberries hang down from the hull, not up, as it may be seen in [17], and at most that, in terms of graphs (the fact that the two options are so different should already, for any primary goer in Mathematics, exclude the English word from the language of science, for science has got, as most fundamental objective, making people from everywhere on Earth see the same image, when given a name, as the image envisaged by the person who ∗

Postal address: PO Box 12396, A’Beckett st, Melbourne, Victoria, Australia, 8006. Electronic address: [email protected].

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has coined that expression). This paper is organized as follows: • The so many definitions for hull: Presentation and analysis; • Proposal of axioms of definition for both ‘calyx’ and ‘holder’ (addition to the mathematical lingo); • Fixing of old definitions, those which bear any ethical sense in Mathematics, as to what should be, according to the well-posedness theory for Mathematics; • Conclusion; • References.

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The so many different definitions for hull

2.1

Definition number 1 for hull

In [7], a polynomially convex hull of a compact subset of the complex plane, which they call K, is: b = {z ∈ C : |p(z)| ≤ maxζ∈k |p(ζ)| for all polynomials p}. K In the same source, we find this remark: A compact set K is told to be b polynomially convex if K = K. 2.1.1

Analysis of the definition number 1 for hull

-Theory needed: • Polynomial expression ([2]): A polynomial in x is an algebraic expression that may1 be written in the form an xn + an−1 xn−1 + ... + a1 x + ao , where n is a nonnegative integer, x is a variable, and each one of the numbers ao , a1 , ..., an are called the coefficients of the polynomials. If an 6= 0, we say this is a polynomial expression of degree n, an is known as the leading coefficient, and ao is called the constant coefficient. • Polynomial function ([2]): Any function f that has the form f (x) = an xn + an−1 xn−1 + ... + a1 x + ao , where n is a nonnegative integer and 1

we have replaced ‘can’ with ‘may’ when reproducing the extract

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ao , a1 , ..., an are real numbers is called polynomial function. If an 6= 0 we say that f is a polynomial function of degree n, an is known as the leading coefficient, and ao is called the constant coefficient. - Geometric shape intended: We consider the modulus of the square of z as our best example, found mentioned in the page 57 of [18]. Notice that, in this case, it is very clear that taking all couples of complex numbers fulfilling the demands of the polynomial function leads us to a shape which is similar (pushing it a bit?) to an inverted strawberry hull for the part above the horizontal axis, with the set called ‘hull’ keeping on going after that axis, going down the same amount of length it went up, trivially. Another possible interpretation would be that the set K, after going through the operation of ‘hull formation’, would return a convex complex slice. Considering only the family of polynomial functions of the sort p(z) = a1 z 2 , we notice that, once a1 ∈ <, we may choose a1 as close to zero as we wish, b may actually reduce to a point. See: so that K b z ∈ K ⇐⇒ z ∈ S, where S = all collections formed from z ∈ C/|pm (z)| ≤ maxζ∈k |pm (ζ)|, where m varies from 1 to sup Infta([10]. Here, not mattering our choice of K, we may always choose a suitable a1 such that we always b end up with a point as K). - Criticisms: • the graphical concept of ‘hull’ does not remind us of either a strawberry top or a case; • ‘Hull’ is not a univocal sigmatoid, therefore it should not be even considered as a candidate to pertinence to the mathematical lingo; • The expression ‘polynomially convex’ should refer to at least two mathematical concepts: Convexity and polynomials. Due to the operations with complex numbers, we will end up with a continuous region being the ‘hull’. Such a region will not contain holes, therefore, or any two points which may be joined giving us a line outside of the region. b everything is Because we have used polynomials to reach the set K, harmonious. Probably the case that the word hull should be replaced with the word intersection, however. • They clearly meant polynomial function instead of polynomial because we are after an evaluation of the polynomial over a domain point there. 3

2.1.2

Final diagnosis for the definition 1 for hull

If we take away the word ‘hull’ and replace ‘polynomial’ with ‘polynomial function’, we may be starting to change the definition 1 into something more ‘acceptable’ in Mathematics.

2.2

Definition 2 for hull

In [8], we find the following definition: For every E ⊂ V , the intersection ch(E), of all convex sets containing E is a convex set called the convex hull of E, that is: n n X X ch(E) = { λj xj ; λj ≥ 0; λj = 1; xj ∈ E; n = 1, 2, ...}. 0

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Remark 1. One will also find the notation ‘co e’ for convex hull. 2.2.1

Analysis of the definition 2 for hull

- Theory needed: • From [8]: Definition 3. Let V be a vector space over <. A subset X ⊂ V is called convex if every line intersects X in an interval, that is: (λx1 + (1 − λ)x2 ) ∈ X when 0 < λ < 1 and x1 , x2 ∈ X. Therefore, the empty set is a convex set because there are no elements from which to form lines, what makes the inclusion condition be trivially verified. Boring enough, [15], page 108, brings us the certainty that any intersection of convex sets is a convex set. It is straight forward that the smallest convex set containing a set can only be the set built from all linear combinations between each and every couple of its elements. - Geometric shape intended: With the definition 2, the geometric shape intended is the connection, by couples, in an exhaustive manner, of all the points in the original set. - Criticisms:

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• A definition, in Mathematics, that is, a creation axiom, should be written obeying the well-posedness guidelines. That would mean lowest number of words as possible, unique referents for world objects, or closest as possible to that, most objective wording as possible, but still complete in the sense that any reader would be able to form the same abstract idea (or even concrete) as that intended by the creator of that axiom when reading it. An axiom is something unwanted in Mathematics for it has got no proof. Therefore, its creation must be justified with argumentation. The worst mistake of all in definition 2 is, therefore, the inclusion of the information ‘the intersection of all convex sets containing E, is itself a convex set’ (the line is redundant and refers to a theorem, therefore something provable, not suitable to be included in an axiom); • Hull is not a good candidate for inclusion in the mathematical lingo, as explained before, and it should be immediately replaced by a word that satisfies the requirements for inclusion in the mathematical vocabulary. The most obvious candidate is ‘smallest convex set containing a set’. It is also possible to find the expression ‘convex cone’ describing the same sort of entity (see [13], for instance). Cone is a mathematical referent, supposed to uniquely refer to a geometric shape, therefore is not a term that is suitable for our purposes either; • The other point, which could be made, regards possible redundancy of the own concept. It seems obvious that if we group all convex combinations of the elements of a set in another set then the new set, formed this way, will be the smallest convex set containing the elements of the set. The concept seems to fit better in a remark, following the definition axiom for convex sets, than anything else. 3.0.2

Final diagnosis for the definition 2, hull

The definition seems to be redundant.

3.1

Definition 3 for hull

The convex hull of any set S is the union of the convex spans of all the finite subsets of S.

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3.2

Analysis of the definition 3 for hull

- Theory needed: In the Macquaire dictionary, the sigmatoid ‘span’ is found defined as: “ ... 10. to extend over or across 11. to provide with something that extends over: to span a river with a bridge 12. to extend, reach, or pass over (space or time)... ([10])”. - Geometric shape intended: A convex ‘cover’, or ‘envelope’, for the original set S. - Criticisms: • The word ‘hull’ is inappropriate, as explained before, to designate our concept; • The ‘finite subsets’ is a cloudy and dangerous part of definition 3. First of all, the real numbers form an infinite set, convex one, for it is not possible to join two points and get a ‘hole’ in doing so. Yet, with finite subsets, one will never cover it, so that it would never have a convex hull and that does not seem to be the intentions of the ‘trial of definition axiom’. 3.2.1

Final diagnosis for the definition 3, hull

This definition looks more reasonable in its shape, but it is far more equivocated in its wording. Best if entirely forgotten.

3.3

Definition 4 for hull

In [3], we find the following definition: On an Euclidean space, of even dimension, we may introduce, by a choice of complex valued coordinate functions z1 , z2 , ..., zn , the structure of the complex n−space, C n . We can then associate, with each compact subset X, of our space, the polynomial convex hull, in C n , denoted by hull(X). By definition, hull(X) is the set of all p in C n , which satisfy the relation |f (p)| ≤ maxx∈X |f (x)|, for every polynomial f (z, ..., zn ). When X = hull(X), we say that X is polynomially convex in C n . 6

3.3.1

Analysis of the definition 4 for hull

- Theory needed: • An Euclidean space bears complete metrics, that is, all foundational necessary axioms for a metrics to deserve being called a ‘metrics’, according to Euclid, are valid there. Basically, it is the place where Euclid would feel comfortable at (his axioms and theorems). That means one is able to calculate distance and work with the notion, all the way through, with no further problems. Dimension, in Mathematics, is equivalent to coordinate bearer universe. A compact set, which is complex, means the usual: Limited and closed. However, in the reals, it is an interval, easy to deal with. In the complex plane, it has to be described graphically, or by means of a reference to the ‘real’ part of its members. • A function, in Mathematics, is usually referred to by means of ‘f’, whilst a polynomial gets referred to by means of ‘p’, that is, we make it easy and quick so that any reader will remember the first letter of the words, what then works even in a psycholinguistics level. Therefore, when no need arise, when the function, or polynomial, being written about is only one, any mathematician has got, as minimum ethical obligation (help fellows to understand and progress further on nonreserved work), to use those letters in their scientific statements. This is a silent rule for Mathematics, made via habit and history, rather than via statements from its theory. - Geometric shape intended: None. -Criticisms: • The word ‘hull’ is inadequate, as already explained; • We cannot ‘introduce’ the structure of something that already exists in Mathematics. We may, at most, apply that structure, perhaps replacing the usual designations for the coordinates with other designations; • A decision must be made, in what regards the entities we refer to: Either polynomial or functions (the definition 1 is similar to this one and the entity there is polynomial); 7

• If we choose functions, as it appears to be a better choice (see analysis of the definition 1), a better description of what we intend would perhaps be ‘functionally convex’ rather than ‘polynomially convex’; • A function might not be continuous and we can only guarantee compacts are taken into compacts in this special case. Perhaps we should add the word polynomial, or a short mathematical description of the polynomial form for the function, to guarantee we hold a maximum there, therefore; • It is necessary to determine the counter-domain for the function, and this might be C, instead of C n . However, before a ‘shape’, like a geometrical shape, is intended with the concept, it seems a bit odd to call it ‘convex’ something. 3.3.2

Final diagnosis for the definition 4, hull

Perhaps it may be saved. It should definitely be considered in association with the definition 1, and the concepts should be seriously related.

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New born babies: Holder and Calyx

We wish for something that is born in the origin of the ‘World of Infinita’ (please refer to [10] for this matter) and dies by the own curve representing the function. This way, we believe not only we get the idea of ‘bouquet’, making it all beautiful, but we also get the idea of ‘support’ for the ‘line of the function’. Besides, not to state this is the most optimized way of making it be, we also get a path to pursue movement of randomly chosen image points, being able to finally work out, by hand, the closest convex function curve, for instance, which is available. We are then interested in a straight line born at the origin of the W.I., which is going to die at the image of the point (one for each function domain point and attached to it in an explicit manner). We get the names ‘holder’ and ‘calyx’ for finding them suitable for inclusion in the mathematical lingo (both point uniquely to references in the actual world: One usually associated with things that will ‘hold’ the weight of something else, like balloons and etc., what makes of it an ideal thing to think of (psycholinguistics), ‘holding the function in the air’, other usually associated with the situation of the strawberry (as from the plant) hull - see [4] and [1], for instance).

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This way, we have:

4.1

Holder of a function

In the real numbers environment: Definition 5. Holder of a function f , which we represent by Holder(f(x)), is defined as Holder(f (x)) = {gi (x) ∈ X : gi (x) =

yi x, if (xi 6= 0 ∧ yi 6= 0) v gi (x) = nx xi

( n as big as necessary so that we have the y axis, effectively ) if xi = 0 v gi (x) = 0, 0 ≤ x ≤ xi if (yi = 0 ∧ xi ≥ 0) or xi ≤ x ≤ 0 if (xi < 0 ∧ yi = 0) /(xi ; yi ) ∈ f ∧ i ∈ N }, where X is the set of functions which either start or stop at the origin of the W.I. and hold as other extreme the ordered pair (xi ; yi ), that is: 0 ≤ gi (x) ≤ f (xi ) if f (xi ) ≥ 0 and 0 ≥ gi (x) ≥ f (xi ) if f (xi ) < 0.

5.1

Calyx of a function

The idea of calyx has come from trying to understand the idea of hull. what has been imagined is the region under and over the graph of a function. It will obviously not work for concave functions or convex graphs. We then wish to restrict the scope of functions, which are passive of having a calyx associated to them, to those with concave curves as representation or their alike in spaces of more dimensions.

5.2

Short calyx - real numbers environment

Consider K, a compact real interval. Consider f , a polynomial non-negative function, which determines a concave curve when evaluated over K. Under the just mentioned conditions, SCf is the result of a process: (x; yn ) ∈ SCf ⇐⇒ 0 ≤ yn ≤ f (x), x ∈ K, n ∈ N.

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5.2.1

Long calyx: Real numbers environment

Consider K, a compact real interval. Consider f , a polynomial non-negative function, which determines a concave curve when evaluated over k. Under the just mentioned conditions, LCf is the result of a process: (x; yn ) ∈ LCf ⇐⇒ yn ≤ f (x), x ∈ K, n ∈ N.

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Conclusions

In this paper, we have discussed the idea of hull in depth, in a depth that has never been found in the Mathematics literature so far. We have discussed the varied definitions associated with the term and we have come up with several criticisms to each one of them, being unable to propose any new definition for simply finding the term ‘hull’ inadequate for Mathematics. Besides, the proposed mathematical definitions, which would refer to hull, seem to be either redundant or improper. On the way, however, we have proposed the idea of calyx of a function and the idea of holder of a function, with respective univocal definitions, univocal also in what regards their geometry.

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References

[1] Calyx. (2008). In Merriam-webster Online Dictionary. Retrieved May 13th, 2008, at http : //www.merriam − webster.com/dictionary/calyx. [2] D. A. Smith Library of Math-Online math organized by subject into topics. Found online at http : //www.libraryof math.com/about − us.html since 2005, as seen on the 5th of May of 2008. [3] G. Stolzenberg. Polymially and rationally convex sets. Acta Mathematics, issue volume 109, nr.1, July, 1963, pp. 259-289. Found online at http : //www.springerlink.com/content/v521123483860h73/f ulltext.pdf, accessed on the 6th of May of 2008. [4] Holder. (2008). In Merriam-webster Online Dictionary. Retrieved on the 13th of May of 2008 from http : //www.merriam − webster.com/dictionary/holder. [5] Hull. (n.d.) Collins Essential Thesaurus 2nd edition. (2005,2006). Retrieved on the 10th of May of 2008 from http : //www.merriam − webster.com/dictionary/hull. [6] Hull. (2008). In Merriam-webster Online Dictionary. Retrieved on the 11th of May of 2008 from http : //www.merriam − webster.com/dictionary/hull. [7] J. B. Conway. Functions of one complex variable 1. Springer, 1975. ISBN: 0387903283. [8] M. R. Pinheiro. Convexity Secrets. Trafford Publishing. 2008. ISBN-10: 1425138217. [9] M. R. Pinheiro. Exploring the concept of S-convexity. Aequationes Mathematicae, v. 74, 3(2007). [10] M. R. Pinheiro. Infinis que seras tamen. Online preprint located at www.pdfcoke.com/illmrpinheiro2. 11

[11] Macquaire dictionary. Macquaire Dictionary Publishers Pty Ltd, p. 1577, 5th ed., 2009, ISBN : 9781876429669(hbk.). [12] M. R. Pinheiro. S-convexity (Foundations for analysis). Differential Geometry-Dynamical systems, vol. 10, no. 1, 2008. [13] M. R. Pinheiro. S-convexity revisited (long version. Optimization Letters, DOI 10.1007/511590-008-0087-4, 2008. [14] M. R. Pinheiro. The inferential step in the Sorites Paradox: Logical or human? Online preprint located at www.pdfcoke.com/illmrpinheiro2, 2005. [15] N.K. Nikolskii. Functional Analysis. Springer, 1992. ISBN: 3540505849. [16] S. Lang. Functions of complex variables. Springer, 1993. ISBN: 0387985921. [17] Strawberry plant stock photos and images (2008). Foto search stock photography and stock footage royalty free image. Retrieved on the 11th of May of 2008. http : //www.f otosearch.com/photos − images/strawberry − plant.html [18] T. Needham. Functions of complex variables. Oxford University Press, 1999. ISBN: 0198534469.

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