Geometry of Polynomial Spaces and Polynomial Inequalities”

IX Escuela-Taller de Análisis Funcional (2019)

Geom etry of polynom ialspaces and polynom ial

inequalities

M anuelBernardino A b s t r a c t : This paper summarises the contents of the course on geometry of polyno- Universidad de Sevilla mial spaces and polynomial inequalities delivered at the 9th workshop of Functional [email protected] Analysis organised by the Spanish Functional Analysis Network in Bilbao between the 3rd and the 8th of March, 2019, in memoriam of Prof. Bernardo Cascales. We

L u i s G ó m e z first survey the most relevant results needed to understand polynomials innormed Universidad de Extermadura spaces. Then, we provide a few examples of polynomial spaces whose extreme [email protected] points are fully described, and a couple of applications of the so-called Krein- Milman approach to obtain several sharp polynomial inequalities.

Estrella Lavado

Universidad de Extermadura R e s u m e n : Este artículo resume el contenido del curso sobre geometría de espacios [email protected] polinomiales y desigualdades polinomiales que tuvo lugar en la IX Escuela-Taller de Análisis Funcional organizada por la Red de Análisis Funcional y Aplicaciones

Clara M artínez en Bilbao entre el 3 y el 8 de marzo de 2019, in memoriam al profesor Bernardo Universitat de València Cascales. Repasamos los resultados más relevantes para entender los polinomios [email protected] en espacios normados. Después proporcionamos algunos ejemplos de espacios polinomiales cuyos puntos extremos están completamente descritos y un par de aplicaciones del llamado método de Klein-Milman para obtener desigualdades Souleym ane Ndiaye polinomiales óptimas. Universidad de Murcia [email protected] K e y w o r d s : Bernstein and Markov inequalities, unconditional constants,

Nuria Storch polarizations constants, polynomial inequalities, homogeneous polynomials, Universidad Complutense de Madrid extreme points.

[email protected] M S C 2 0 1 0 : 46G25, 46B28, 41A44.

Course instructor

� Gustavo M uñoz Universidad Complutense de Madrid [email protected]

R e f e r e n c e : BERNARDINO, Manuel; GÓMEZ, Luis; LAVADO, Estrella; MARTÍNEZ, Clara; NDIAYE, Souleymane; STORCH, Nuria, and MUÑOZ, Gustavo. “Geometry of polynomial spaces and polynomial inequalities”. In: TEMat monográficos, 1 (2020): Artículos de las VIII y IX Escuela-Taller de Análisis Funcional, pp. 79-96. ISSN: 2660- 6003. URL: https://temat.es/monograficos/article/view/vol1-p79.

cb This work is distributed under a Creative Commons Attribution 4.0 International licence https://creativecommons.org/licenses/by/4.0/ Geometry of polynomial spaces and polynomial inequalities

1 . Introduction

This paper contains three main blocks. The first one is devoted to introducing polynomials in normed spaces. Although polynomials in a finite number of variables are well known to all undergraduate students, itisnot so clear what polynomials in infinitely many variables are. In this block we will present all definitions and basic results required to understand polynomials in an arbitrary normed space, including the polarisation formula and polarisation constants. The second block deals with the geometry of polynomial spaces. The reader is referred to Dineen’s book [15] for a modern monograph on polynomials on normed spaces. The characterisation of the extreme points of polynomial spaces is a question that has called the interest of a significant number of researchers in the last decades [1, 5, 10, 11, 18, 22, 24, 28–31, 33, 34]. This problem conveys a tremendous difficulty in most infinite (or even finite) dimensional polynomial spaces of interest, but in very specific cases, a complete and explicit description of the extreme points can be given.Wewill focus on a number of these particular examples, providing the reader not only with the extreme points of several -dimensional polynomial spaces, but also with a formula to calculate the polynomial norm, a parametrisation of the unit sphere and nice pictures of the unit balls of those spaces. Finally, a third section contains3 the applications of the geometrical results of the second block. In order to understand the applications, a precise introduction to several well-known polynomial inequalities will be provided. It is possible to find a vast diversity of applications in the literature, and therefore we will make a (restrictive) selection consisting of two types of polynomial inequalities, namely, the polynomial Bohnenblust-Hille inequality and the Bernstein-Markov type inequalities. The arrangement described in the previous paragraph respects the structure of the course on geometryof polynomial spaces and applications delivered during the 9th workshop of Functional Analysis organized by the Spanish Functional Analysis Network in Bilbao between the 3rd and the 8th of March, 2019, in memoriam of Prof. Bernardo Cascales.

2 . Polynom ials in norm ed spaces

In this section we present the essential definitions and results needed to understand polynomials in normed spaces. We begin by recalling a number of basic concepts and definitions related to polynomials in a finite number of variables. In order to handle monomials in 푛, we introduce the following notation. An 푛-dimensional multiindex is an 푛-tuple 훼 = (훼 … 훼푛) with 훼푖 ∈ ℕ ∪ { } for all 푖 = … 푛. If 푥 = 푛 훼 (푥 … 푥푛) ∈ , then |훼|, 훼 and 푥 represent, respectively, 1 , , 훼 훼푛 0 1, , 1 훼 + … + 훼 훼 ⋯ 훼 and 푥 ⋯ 푥푛 . , , ! 푛 푛 1 푚 With the above notation, a polynomial1 in of1 degree at most 푛 is a1 linear combination of monomials of , 푚 ! ! the form 푥훼 with 푥 ∈ 푚 and 훼 ∈ (ℕ ∪ { }) with |훼| ≤ 푛. Hence, a polynomial 푃 in 푚 variables (real or complex) of degree at most 푛 has the form

0 훼 푚 푃(푥) = ∑ 푎훼푥 for all 푥 ∈ 훼∈(ℕ∪{ })푚 |훼|≤푛 0 , , with 푎훼 ∈ . Accordingly, a polynomial 푃 in 푚 variables is homogeneous of degree 푛 if

훼 푚 푃(푥) = ∑ 푎훼푥 for all 푥 ∈ 훼∈(ℕ∪{ })푚 |훼|=푛 0 , , with 푎훼 ∈ . Our first objective is to recall how to extend the above well-known definition of polynomial and homogeneous polynomial to arbitrary linear spaces. 푛 Let 퐸 be a linear space over ( = ℝ or ℂ). From now on, for each 푛 ∈ ℕ, ℒ푎 ( 퐸) denotes the space of all 푛-linear forms on 퐸. Recall that 퐿 is 푛-linear if it is linear in every coordinate. As usual, 퐸푛 ≔ 퐸 × 푛)… × 퐸. Also, we consider the diagonal mapping

푛 훥푛 ∶ 퐸 → 퐸 푥 ↦ (푥 … 푥).

80 , , https://temat.es/monograficos Bernardino et al.

D efinition 1 (homogeneous polynomials). If 퐸 is a linear space over and 푛 ∈ ℕ, we say that 푃 is an 푛 풏-homogeneous polynomial if there exists 퐿 ∈ ℒ푎 ( 퐸) with 푃 = 퐿 ∘ 훥푛. Equivalently, we write 푃 = 퐿.̂ 푛 The space of all 푛-homogeneous polynomials on 퐸 is denoted by 풫푎( 퐸). We say that 푃 is a polynomial of 푘 degree at most 푛 on 퐸 if 푃 = 푃푛 + … + 푃 + 푃 , where 푃푘 ∈ 풫푎( 퐸) for 푘 = … 푛 and 푃 is a constant. ◀ 푛 푛 Observe that, if 푃 ∈ 풫푎( 퐸), then 푃 (휆푥1 ) =0 휆 푃 (푥) for all 푥 ∈ 퐸 and every 휆 ∈ .0 This property is also satisfied by all homogeneous polynomials in a finite number of1, variables., On the other hand,the 푛-linear form that defines a given 푛-homogeneous polynomial is not uniquely determined. Let us see this with an example. First, notice that for all bilinear forms 퐿 on 푛, where 푛 ∈ ℕ, there exists an 푛 × 푛 matrix with entries in such that 퐿 (푧 푤) = 푧퐴푤> for all 푧 푤 ∈ 푛. Here 푤> means, as usual, the transpose of 푤. Hence, all -homogeneous polynomials on 푛 are of the form , , 푛 2 푃 (푧) = 퐿 (푧 푧) = ∑ 푎푖푗푧푖푧푗 푖 푗=

푛 , > ,> > 푛 where 퐴 = (푎푖푗)푖 푗= . If we consider now 퐵 = (퐴 + 퐴 ), then1 푧퐴푧 = 푧퐵푧 for every 푧 ∈ . The latter means that the two bilinear forms determined1 by the matrices 퐴 and 퐵 define the polynomial 푃. , 1 The previous example motivates the definition2 of symmetric multilinear forms.

D efinition 2 (symmetric multilinear forms). Let 퐸 be a linear space over and let 푛 ∈ ℕ. An 푛-linear form 퐿 is symmetric if 퐿 (푥 … 푥푛) = 퐿 (푥휍( ) … 푥휍(푛)) for all (푥 … 푥 ) ∈ 퐸푛 and every permutation 휎 of { … 푛}. The space of all symmetric 푛-linear forms on 푛 1 1 푠 푛 퐸 is denoted by ℒ푎 ( 퐸). , , , , ◀ 1 , , 1,푛 , 푠 푛 R e m a r k 3 . Let us consider the mapping 푠 from ℒ푎 ( 퐸) onto ℒ푎 ( 퐸) given by

푠 (퐿)(푥 … 푥 ) = ∑ 퐿 (푥 … 푥 ) 푛 푛 휍( ) 휍(푛) 휍∈푆푛 1 1 1 where 푆푛 is the group of all permutations, of, { … 푛}. Then, 푠 is a projection., , , Furthermore, 퐿 and 푠(퐿) define ! 푛 푠 푛 the same homogeneous polynomial. Therefore, if 푃 ∈ 풫푎( 퐸), it is always possible to choose 퐿 ∈ ℒ푎( 퐸) such that 퐿̂ = 푃. 1, , ◀ Now, using the so-called multinomial formula, it is possible to see that definition 1 does extend the concept of homogeneous polynomial from a finite number of variables to arbitrary linear spaces.

푛 Proposition 4 (multinomial formula). Let 퐸 be a real or complex linear space, 푃 ∈ 풫푎 ( 퐸), 푥 … 푥푘 ∈ 퐸 and 푎 … 푎푘 ∈ . Then, 1 푘 , , 1 푛 , , 푃 (∑ 푎 푥 ) = ∑ 푎푚 ⋯ 푎푚푘퐿 (푥푚 … 푥푚푘) 푖 푖 푚 푘 푘 푖= 푚∈(ℕ∪{ })푘 1 1 |푚|=푛 ! 1 1 1 , , , 푠 푛 0 ! where 퐿 ∈ ℒ푎( 퐸) satisfies 퐿̂ = 푃 and

푚 푚푘 퐿 (푥푚 … 푥푚푘) ≔ 퐿( ⏞⎴⏞⎴⏞푥 … 푥 … ⏞⎴⏞⎴⏞푥 … 푥 ). 푘 1 푘 푘 1 1 푘 1 1 P r o o f . Let 푚 = (푚 … 푚푘) ∈ { …, 푘}, with |푚| = 푛, and, define, , , , 푘 퐴 = {(푖1 … 푖 ) ∈ { … 푘} such that appears 푚 times … 푘 appears 푚 times} 푚 , , 푘 0, 1, , 푘 and 1 1 , , 0, 1, , ∑ 1 , , 푎푚 = 퐿(푥푖 … 푥푖푘). (푖 … 푖푘)∈퐴푚 1 Then, using linearity, 1, , , , 푛 푚 푚푘 푃(푎 푥 + … + 푎푘푥푘) = 퐿((푎 푥 + … + 푎푘푥푘) ) = ∑ 푎 ⋯ 푎푘 푎푚. 푚∈(ℕ∪{ })푘 1 1 1 1 1 |푚|=푛 1 0 TEMat monogr., 1 (2020) e-ISSN: 2660-6003 81 Geometry of polynomial spaces and polynomial inequalities

Observe that, due to the symmetry of 퐿, we have

푚 푚푘 푎푚 = 퐿(푥 … 푥푘 ) ⋅ #(퐴푚) 1 1 푛 where #(퐴푚) denotes the cardinality of 퐴푚. Using elementary combinatorics we arrive at #(퐴푚) = , , , , 푚 ⋯푚푘 which concludes the proof. ! ▪ 1! ! The importance of the multinomial formula in this context relies on the fact that it can be usedtoextend the classical definition of polynomials in several variables. Indeed, let 퐸 be a real or complex linear space and 퐹 a finite dimensional subspace of 퐸. Let {푒 … 푒푘} be a Hammel basis for 퐹 and 푥 = 푥 푒 +…+푥푘푒푘 ∈ 퐹. Then, using the multinomial formula, 1 1 1 , , 푃 (푥) = 푃 (푥 푒 + ⋯ + 푥푘푒푘)

푛 푚 푚푘 푚 푚푘 푚 = ∑1 1 푥 ⋯ 푥 퐿 (푒 … 푒 ) = ∑ 푎 푥 푚 푘 푘 푚 푚∈(ℕ∪{ })푘 1 1 푚∈(ℕ∪{ })푘 |푚|=푛 ! 1 1 |푚|=푛 , , , 0 ! 0 which shows that the restriction of an 푛-homogeneous polynomial in the sense of definition 1 to a 푘-dimensional space is an 푛-homogeneous polynomial in 푘 variables.

2 . 1 . The polarization form ula

푛 푠 푛 If 퐸 is a linear space and 푃 ∈ 풫푎( 퐸), we have seen in remark 3 that there exists 퐿 ∈ ℒ푎( 퐸) such that 푃 = 퐿.̂ We can go further and prove that this symmetric 푛-linear form that determines 푃 is unique, and we call it polar of 푃. The polarization formula does not only prove the uniqueness of the symmetric 푛-linear form that defines a given 푛-homogeneous polynomial: additionally, it provides an explicit expression of the polar in terms of the polynomial it defines. There are many forms of the polarization formula. Wehave chosen one taken from Dineen’s book [15] that uses Rademacher functions.

푛 푠 푛 T h e o r e m 5 (polarization formula). Let 퐸 be a real or complex linear space, 푃 ∈ 풫푎 ( 퐸), 퐿 ∈ ℒ푎 ( 퐸) and 푛 assume that 퐿̂ = 푃. If (푥 … 푥푛) ∈ 퐸 , then

1 , , ( 1 ) 퐿 (푥 … 푥푛) = ∫ 푟 (푡) ⋯ 푟푛 (푡) 푃 (푟 (푡) 푥 + … + 푟푛 (푡) 푥푛) d푡 푛 1 1 1 1 1 1 where 푟 is the 푗-th Rademacher, , function, defined by , 푗 ! 0

푗 푟푗 (푡) = sign(sin( π푡)) for all ≤ 푗 ≤ 푛.

푛 P r o o f . Let 푚 = (푚 … 푚 ) ∈ (ℕ ∪ { }) . Recall that the Rademacher functions satisfy 푛 2 1

1 푚 + 푚푛+ 푚 + 푚푛+ ( 2 ) , ,∫ 푟 (푡) ⋯0 푟푛 (푡) d푡 = ∫ 푟 (푡) d푡 ⋯ ∫ 푟푛 (푡) d푡 1 1 1 1 1 1 1 1 1 1 1 and 0 0 0

푚푗+ if 푚푗 + is even, ( 3 ) ∫ 푟푗 (푡) d푡 = { 1 if 푚 + is odd. 1 푗 1 1 0 푚 + 0 푚푛+ 1 If we assume |푚| = 푛, the product ∫ 푟 (푡) d푡 ⋯ ∫ 푟푛 (푡) d푡 vanishes unless 푚 = … = 푚푛 = , in 1 1 1 1 1 1 1 0 0 1

82 https://temat.es/monograficos Bernardino et al. which case its value is . Now, using the multinomial formula,

∫ 푟 (푡)1⋯ 푟푛 (푡) 푃 (푟 (푡) 푥 + … + 푟푛 (푡) 푥푛) d푡 1 1 1 1 0 = ∫ 푟 (푡) ⋯ 푟푛 (푡) 퐿 (푟 (푡) 푥 + … + 푟푛 (푡) 푥푛 … 푟 (푡) 푥 + … + 푟푛 (푡) 푥푛) d푡 1 1 1 1 1 1 푛 푚 푚,푛 , = ∫0 푟 (푡) ⋯ 푟푛 (푡) ∑ 푟 (푡) ⋯ 푟푛 (푡) 퐿 (푥 … 푥푛) d푡 1 푛 푚 푚∈(ℕ∪{ }) 1 1 |푚|=푛 ! 1 1 0 0 , , 푛 푚 + ! 푚푛+ = ∑ ∫ 푟 (푡) ⋯ 푟푛 (푡) d푡퐿 (푥 … 푥푛) = 푛 퐿(푥 … 푥푛). ▪ ℕ 푛 푚 1 푚∈( ∪{ }) 1 1 1 |푚|=푛 ! 1 1 1 푛 0 0푠 푛 , , ! , , R e m a r k 6 . Let 푃 ∈ 풫푎( 퐸) and 퐿 ∈! ℒ푎( 퐸) be the polar of 푃. It has already been mentioned that there are many forms of the polarization formula. To see this we just need to replace the Rademacher functions by any set of functions satisfying the identities (2) and (3). For instance, we may consider any set of 푛 independent and orthonormal random variables 푟 … 푟푛 on [ ] taking values on and 푛 1 훹 = ̄푟 ⋯푛 ̄푟, ⋅ 푃, (∑ 푟푖푥0,푖) 1. 푖= 1 Proceeding as in the proof of theorem 5, the expectancy of1 훹 would be given by

[훹] = 푛 퐿(푥 … 푥푛) that is, 1 ! , , , 퐿(푥 … 푥 ) = [훹] 푛 푛 which is a much more general way to express the1 polarization1 formula (1). Thus, if 푟 … 푟푛 are 푛 independent , , , Bernouilli random variables taking the value − with probability! / and with probability / , we would 1 have , , 1 1 푛2 1 1 2 ( 4 ) 퐿 (푥 … 푥 ) = ∑ 휀 ⋯ 휀 푃 (∑ 휀 푥 ). 푛 푛푛 푛 푖 푖 휀푖=± 푖= 1 1 1 This is a very convenient form, to, put down the polarization formula. ◀ 2 ! 1 1

2 . 2 . Continuity in polynom ialspaces

All polynomials in finitely many variables are continuous. However, this is far from being truewhen polynomials on an infinite dimensional normed space are considered. Actually, continuity fails tobe universal even for linear forms on an infinite dimensional normed space. Let (퐸 ‖⋅‖) be a normed space over . We represent the space of continuous 푛-homogeneous polynomials, the space of continuous 푛-linear forms and the space of continuous symmetric 푛-linear forms, respectively, 푛 푛 푠 푛 푛 푠 푛 by 풫 (, 퐸), ℒ ( 퐸) and ℒ ( 퐸). Also, for 푃 ∈ 풫( 퐸) and 퐿 ∈ ℒ ( 퐸), we define ‖푃‖ = sup {|푃 (푥)| ∶ ‖푥‖ ≤ }

‖퐿‖ = sup {|퐿 (푥 … 푥푛)| ∶ ‖(푥 … 푥푛)‖ ≤ } 1 , where 1 1 , , , , 1 , ‖(푥 … 푥푛)‖ = sup {‖푥푖‖ ∶ ≤ 푖 ≤ 푛} . 푛 푛 These definitions are intended to1 introduce a normin 풫( 퐸) and ℒ( 퐸). However, at this stage we do not even know whether ‖푃‖ or ‖퐿‖ are finite., , As a matter of1 fact, the finiteness of ‖푃‖ or ‖퐿‖ characterizes the continuity of 푃 or 퐿, as we will see later. First we present a fundamental result in the theory of polynomials in normed spaces also known as the polarization inequality. The proof provided below is taken from Dineen’s book [15].

TEMat monogr., 1 (2020) e-ISSN: 2660-6003 83 Geometry of polynomial spaces and polynomial inequalities

T h e o r e m 7 (polarization inequality). Let 퐸 be a normed space. Then, 푛푛 ‖푃‖ ≤ ‖퐿‖ ≤ ‖푃‖ 푛 for every 퐿 ∈ ℒ푠 (푛퐸) and 푃 ∈ 풫 (푛퐸) such that 퐿 is the polar of 푃. ! P r o o f . The first inequality is trivial since 푃 is a restriction of 퐿. To prove the second inequality, we use the polarization formula (4):

‖퐿‖ = sup {|퐿 (푥 … 푥푛)| ∶ ‖(푥 … 푥푛)‖ ≤ } | 푛 | = sup {| 1 ∑ 휀 ⋯ 휀 푃1(∑ 휀 푥 )| ∶ ‖(푥 … 푥 )‖ ≤ } | 푛푛 , , 푛 , , 푖 푖 | 1 푛 | 휀푖=± 푖= | 1 | 1 푛 | 1 1 1 , , 1 ≤ 2∑ !sup {|푃 (∑ 휀푖푥푖)| ∶ ‖(푥 … 푥푛)‖ ≤ } 푛푛 | | 휀푖=± 푖= 푛1푛 | 푛 | 1 1 1 , , 1 = 2 ! ∑ sup {|푃 ( ∑ 휀푖푥푖)| ∶ ‖(푥 … 푥푛)‖ ≤ } 푛푛 | 푛 | 휀푖=± 푖= 푛푛 1 1 ≤ ‖푃‖.1 1 , , 1 ▪ 2푛 ! As is well known, boundedness is a characteristic property of continuous linear forms on any normed space. A similar result holds for! homogeneous polynomials, as we are about to see. The proof of the following result is inspired in Dineen’s book [15].

푛 T h e o r e m 8 . Let 퐸 be a normed space over and 푃 ∈ 풫푎 ( 퐸). Then, the following are equivalent:

( i ) 푃 is continuous in 퐸.

( i i ) 푃 is continuous at .

( i i i ) 푃 is bounded over B 퐸, the closed unit ball of 퐸. 0 P r o o f . That (i) implies (ii) is trivial. We prove now that (ii) implies (iii). By continuity at , given 휀 > there exists 훿 > such that |푃 (푥) | < 휀 whenever ‖푥‖ < 훿. Therefore, if 푥 ∈ B 퐸 we have 푥 휀 0 0 |푃(푥)| = |푃 (훿 ⋅ )| = |푃 (훿푥)| < . 0 | 훿2 | 훿푛 훿푛

Thus, 푃 is bounded on B 퐸. 1 Finally, we show that (iii) implies (i). Choose an arbitrary 푥 in 퐸 and take 푥 ∈ 퐸 with ‖푥 − 푥 ‖ ≤ . In particular, ‖푥‖ ≤ + ‖푥 ‖. Then, by Newton’s binomial formula and Martin’s theorem, we have 0 0 1 0 |푃 (푥) − 푃 (푥 )| = |푃 ((푥 − 푥 ) + 푥 ) − 푃(푥 )| 1 |푛− 푛 | 0 | 0 푗 0 푛−푗0 | = |∑ ( )퐿 (푥 (푥 − 푥 ) )| 1 |푗= 푗 | 푛− 0 0 푛 , 푗 푛−푗 ≤ ∑0( ) ‖퐿‖‖푥 ‖ ‖푥 − 푥 ‖ 1 푗= 푗 푛푛 0 푛− 푛 0 ≤ 0 ‖푃‖‖푥 − 푥 ‖ ∑ ( ) ‖푥 ‖푗 푛 1 푗= 푗 푛 0 0 푛 푛 ≤ ( + ‖푥 ‖) ‖푃‖‖0 푥 − 푥 ‖ . 푛!

Recall that ‖푃‖ is finite since 푃 is bounded on B . Without0 loss of generality,0 we may also assume that 퐸 1 ‖푃‖ > . Hence, for an arbitrary 휀 > we can take! 푛 훿 = min { } > 0 0 푛푛( + ‖푥 ‖)푛‖푃‖ ! and we have that |푃(푥) − 푃(푥 )| < 휀 whenever1,‖푥 − 푥 ‖ <0 훿, that is, 푃0is continuous at 푥 . ▪ 1 0 0 0 84 https://temat.es/monograficos Bernardino et al.

R e m a r k 9 . Let 퐸 be a real or complex normed space. Theorems 5 and 8 show two relevant facts:

푛 푠 푛 1 . If 푃 ∈ 풫푎( 퐸) and 퐿 ∈ ℒ푎( 퐸) is its polar, then 푃 is bounded (continuous) if and only if 퐿 is bounded (continuous). 푛 푠 푛 푠 푛 푛 2 . The spaces 풫( 퐸) and ℒ ( 퐸) are topologically isomorphic and ℒ ( 퐸) ∋ 퐿 ↦ 퐿̂ ∈ 풫( 퐸) is a natural isomorphism whose inverse is provided by the polarization formula. ◀

Using the axiom of choice it is easy to construct non-bounded (and therefore non-continuous) polynomials.

E x a m p l e 1 0 . Let 퐸 be any normed space of dimension 픠 (here 픠 is the continuum, or the cardinality of ℝ). 푠 푛 Let ℬ = {푒푥 ∶ 푥 ∈ ℝ} be a Hammel basis of normalized vectors of 퐸 indexed in ℝ and define 퐿 ∈ ℒ푎( 퐸) on ℬ by 퐿(푒푥 … 푒푥푛) = 푥 ⋯ 푥푛. On the rest of 퐸, 퐿 is defined by linearity. Then, the 푛-homogeneous polynomial induced by 퐿 is not bounded on B 퐸, since ℬ ⊂ B 퐸, but 1 1 , , 푛 lim 푃(푒푥) = lim 푥 = ∞. ◀ 푥→∞ 푥→∞

In general, for any normed space 퐸, the algebraic size, measured in terms of dimension, of the set of non- bounded 푛-homogeneous polynomials (respectively non-bounded symmetric 푛-linear forms) is maximal. Consider the sets 풩ℬℒ푠 (푛퐸) and 풩ℬ풫(푛퐸) of, respectively, all the non-bounded symmetric 푛-linear forms and all the non-bounded 푛-homogeneous polynomials on 퐸. Then, Gámez-Merino, Muñoz-Fernández, Pellegrino, and Seoane-Sepúlveda [16] proved in 2012 the following.

푠 푛 T h e o r e m 1 1 . If 푛 ∈ ℕ and 퐸 is a normed space of infinite dimension 휆, then the sets 풩ℬℒ ( 퐸) ∪ { } and 풩ℬ풫(푛퐸) ∪ { } contain a 휆-dimensional subspace. We say then that the sets 풩ℬℒ푠 (푛퐸) and 풩ℬ풫(푛퐸) 휆 are -lineable. 0 0 2 2 2 . 3 . Polarization constants

If 퐸 is a real or complex normed space 푃 ∈ 풫(푛퐸) and 퐿 ∈ ℒ푠(푛퐸) is the polar of 푃, according to Martin’s theorem (theorem 7), 푛푛 ‖퐿‖ ≤ ‖푃‖. 푛 푛푛 The constant 푛 cannot be replaced by a smaller constant in general since equality can be attained for 푛 ! 푛푛 the space 퐸 = ℓ and the polynomial 훷푛(푥 … 푥푛) ≔ 푥 ⋯ 푥푛 and its polar. However, 푛 can indeed be replaced by a smaller! estimate for specific spaces. This motivates the following definition. 1 1 1 , , ! D efinition 12 (polarization constants). If 퐸 is a normed space over , we define the 풏-th polarization constant of 퐸 as 푛 (푛 퐸) ≔ inf{퐾 > ∶ ‖퐿‖ ≤ 퐾‖푃‖ ∀푃 ∈ 풫( 퐸) and 퐿̂ = 푃}. ◀

A somewhat more general; concept than that0 of polarization, constant arises from the following result by Harris [19].

푠 푛 T h e o r e m 1 3 . Let 퐸 be a complex normed space and 푛 … 푛푘 ∈ ℕ. If 푛 = 푛 + ⋯ + 푛푘 and 퐿 ∈ ℒ ( 퐸), then 푛 푛 푛푘 1 푛 ⋯ 푛푘 푛 1 sup{|퐿(푥 … 푥 )| ∶ ‖푥푖‖ = ≤, 푖 ≤, 푘} ≤ ‖퐿‖.̂ 푘 푛푛 ⋯ 푛푛푘푛 1 1 푘 1 !1 ! A similar result with a different, constant, can be1, proved 1 when 퐸 is a real1 normed space. All this serves as a ! motivation for the following definition.

D efinition 14 (generalized polarization constants). Let 퐸 be a normed space over and 푛 푛 … 푛푘 ∈ ℕ with 푛 = 푛 + ⋯ + 푛푘. Then, we define the generalized polarization constant (푛 … 푛푘 퐸) as 1 푛 푛푘 푠 푛 , , , 1 (푛 … 푛 퐸) ≔ inf{푐 ∶ |퐿(푥 … 푥 )| ≤ 푐‖퐿‖̂ ∀퐿 ∈ ℒ ( 퐸) ‖푥 ‖1 = }. ◀ 푘 푘 푖 , , ; 1 1 1 , , ; , , , , 1 TEMat monogr., 1 (2020) e-ISSN: 2660-6003 85 Geometry of polynomial spaces and polynomial inequalities

From theorem 7 we deduce that 푛푛 ≤ (푛 퐸) ≤ 푛 for any normed space 퐸 over , but the exact value of (푛 퐸) for many choices of 퐸 has remained as an 1 ; unresolved problem until today. The calculation of the exact! value of the polarization constants of specific spaces seems to be a challenging problem and yet, significant; progress has been made. Of particular interest are the works of Sarantopoulos [37] and Kirwan, Sarantopoulos, and Tonge [23], where the spaces 푛 푛 satisfying (푛 퐸) = 푛 are studied. At the other end of the scale, according to an old result, if 퐸 is an Euclidean space over , then (푛 퐸) = [2, 21, 38] (see Dineen’s book [15] for a modern exposition). Furthermore, Benítez; and! Sarantopoulos [4] proved that ℝ(푛 퐸) = implies that 퐸 is a real Euclidean space. However, ℂ(푛 퐸) = does not; necessarily1 imply that 퐸 is a complex Euclidean space. The value of (푛 ℓ푝) is known for some choices of 푝 (see for instance [36]),; but most1 of the polarization constants of the classical spaces are; still1 unknown nowadays. For a complete account on polarization constants, we recommend; Rodríguez-Vidanes’s work [35]. The use of the Krein-Milman approach (which will be described right after theorem 15) in combination with a description of the extreme points of certain polynomial spaces may produce good results in the difficult task of calculating polarization constants. The next section is devoted to the study of thegeometry of certain polynomial spaces.

3 . Geom etry ofsom e -dim ensionalpolynom ialspaces

Let 퐸 be a finite dimensional normed space. Recall that 퐶 ⊂ 퐸 is a convex body if it is a compact convex set with nonempty interior. A point 푒3 ∈ 퐶 is an extreme point of 퐶 if it is not an interior point of any segment contained in 퐶. We use the notation ext(퐶) to represent the set of all the extreme points of 퐶. According to the Krein-Milman theorem (or its finite dimensional version proved by Minkowski in -dimensional spaces and by Steinitz for any dimension), the set of the extreme points of a convex body 퐶 in the finite dimensional normed space 퐸 determines 퐶. The precise formulation of this result is3 the following.

T h e o r e m 1 5 (Minkowski-Steinitz). If 퐸 is a finite dimensional normed space and 퐶 ⊂ 퐸 is a convex body, then

( i ) ext(퐶) ≠ ∅.

( i i ) 퐶 = co(ext(퐶)).

Note that co(퐴) is the convex hull of the set 퐴. This result has been used in a large variety of settings to optimize convex functions. In fact, the resultthat allows the optimization is the following:

If 퐶 ⊂ 퐸 is a convex body in the real finite dimensional normed space 퐸 and 푓∶ 퐶 → ℝ is a convex function that attains its maximum in 퐶, then there is a point 푒 ∈ ext(퐶) such that 푓(푒) = min{푓(푥) ∶ 푥 ∈ 퐶}.

We will address to this result as the Krein-Milman approach from now on. A combination of the Krein- Milman approach and an exhaustive description of the extreme points of the unit ball of a polynomial space provides, in many cases, sharp polynomial inequalities. The previous argument motivates the study of the geometry of polynomial spaces. As a matter of fact, many publications have dealt with this question in the past. Konheim and Rivlin [24], as late as in 1966, characterised the extreme points of the space of real polynomials of degree not exceeding 푛, namely 풫푛(ℝ), endowed with the norm

‖푃‖ = sup{|푃(푥)| ∶ 푥 ∈ [− ]}.

Unfortunately, Konheim and Rivlin’s results do not provide an explicit representation of the extreme points 1, 1 of 풫푛(ℝ). Choi, Kim, and Ki [11], on the one hand, and Choi and Kim [10] on the other, characterised the extreme points of 풫( ℓ ) and 풫( ℓ∞). Grecu [18] extended those results to 풫( ℓ푝) for arbitrary 푝 ≥ . 2 2 2 2 2 2 1 86 https://temat.es/monograficos1 Bernardino et al.

Aron and Klimek [1] characterised the extreme points of the space of the real quadratic polynomials on [− ] and the unit disk in ℂ. Muñoz-Fernández and Seoane-Sepúlveda [33] studied the extreme points of the real trinomials on [− ], whereas Neuwirth [34] did the same thing on the unit disk in ℂ. Kim [22] studied1, 1 polynomials on an hexagon or an octagon. Special attention has also been given to polynomials on non-balanced convex1, bodies. 1 Thus, Muñoz-Fernández, Révész, and Seoane-Sepúlveda [31] studied the geometry of the space 풫( 훥) of the -homogeneous polynomials on the simplex 훥 (the triangle of vertices ( ), ( ) and ( )).2 Milev and Naidenov [28, 29] studied the extreme points of the space of polynomials (homogeneous or not) of degree2 at most 2 on 훥. Gámez-Merino, Muñoz-Fernández, Sánchez, and Seoane0,- 0Sepúlveda1, 0 [170,] studied 1 the geometry of the space 풫( ) of the -homogeneous polynomials on the unit square = [ ] . Muñoz-Fernández, Pellegrino, Seoane-Sepúlveda,2 and Weber [30] characterised the extreme points of the2 space 풫( 퐷(훼 훽)) of the -homogeneous polynomials2 on the circular sectors i휃 π π π 퐷(훼 훽) = {푟e ∶ 푟 ∈0, [1 ] 휃 ∈ [훼2 훽]} for 훽 − 훼 = and 훽 − 훼 ≥ π. These results have been generalised in a recent work by Bernal-González,, 2 Muñoz-Fernández,3 Rodríguez-Vidanes, and Seoane- Sepúlveda, [5] for an arbitrary0, 1 , length, of the interval [훼 4훽], .2 , 4 In the rest of this section we will provide a few illustrative examples representing a tiny fraction of the results mentioned above about geometry of polynomial, spaces.

3 . 1 . The geom etry of 풫(ℝ)

Here we consider the space 풫(ℝ) of the real polynomials 푎푥 +푏푥 +푐 of degree not greater than 2, endowed 2 with he norm 2 2 ‖푃‖풫 (ℝ) = sup{|푃(푥)| ∶ |푥| ≤ }.

Elementary calculus shows that 2 1 | 푏 | | 푏 | 푐 | 푏 | | 푎 − 푐| if 푎 ≠ | 푎 | < and 푎 + < ( | 푎 | − ) ‖(푎 푏 푐)‖풫 (ℝ) = { 2 |푎 + 푐| + |푏| otherwise. 1 2 4 2 2 2 2 0, 1 1 1 , The geometry of, this, -dimensional space was investigated by Aron and Klimek [1] (see also the work of Muñoz-Fernández and Seoane-Sepúlveda [33]). The conclusions extracted from these papers are summarised in the following3 result. From now on, if 퐸 is a normed space, B 퐸 and S 퐸 stand for the closed unit ball and the unit sphere of 퐸, respectively. Also, graph(푓) stands for the graph of the function 푓.

T h e o r e m 1 6 . Define 훤(푎) = (√ 푎 − 푎) and 푈 = {(푎 푏) ∈ ℝ ∶ 푎 < and |푏| ≤ min {|푎| 훤(|푎|)}} 2 2 푉 = {(푎 푏) ∈ [−2 ] × [− ] ∶ |푏| ≥ |푎|} , 0 , , 푊 = {(푎 푏) ∈ ℝ 1∶ 푎1 > and |푏| ≤ min {|푎| 훤(|푎|)}} . , , 1, 1 , 2 2 If for every (푎 푏) ∈ ℝ we define 2 , 0 , 2 푓 (푎 푏) = − 푎 − |푏| 푓 (푎 푏) = −푓 (−푎 푏) , + − + and for every (푎 푏) ∈ ℝ with 푎 ≠ we define , 1 , , , , 2 푏 , 푔+(푎 0푏) = − 푔−(푎 푏) = −푔+(−푎 푏) 푎2 then , 1, , , , 4 ( a ) S 풫 (ℝ) = graph (푓+|푊∪푉) ∪ graph (푓−|푈∪푉) ∪ graph (푔+|푊) ∪ graph (푔−|푈).

( b ) B ext2 ( 풫 (ℝ)) = {±(푡 ±훤(푡) − 푡 − 훤(푡)) ∶ 푡 ∈ [ / ]} ∪ {±( )}.

We show a picture2 of S 풫 (ℝ) in figure 1. , , 1 1 2, 2 0, 0, 1 푚 푛 For fixed 푚 > 푛 in ℕ, the2 geometry of the space {푎푥 + 푏푥 + 푐 ∶ 푎 푏 푐 ∈ ℝ} endowed with the sup norm on [− ] has been studied by Muñoz-Fernández and Seoane-Sepúlveda [33] for all possible choices of 푚 푛. The results depend strongly on whether 푚 and 푛 are even or odd., , 1, 1 TEMat, monogr., 1 (2020) e-ISSN: 2660-6003 87 Geometry of polynomial spaces and polynomial inequalities

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F i g u r e 1 : Unit sphere of 풫(ℝ).

2 3 . 2 . The geom etry of 풫( 훥)

Recall first that 풫( 훥) is the space2 of polynomials 푃(푥 푦) = 푎푥 + 푏푦 + 푐푥푦 endowed with the norm defined by 2 2 2 ‖푃‖훥 = sup{|푃(, x)| ∶ x ∈ 훥} where 훥 represents the region enclosed by the triangle in ℝ of vertices ( ), ( ) and ( ) (or simplex, , for short). All the results in this section are taken from the2 work of Muñoz-Fernández, Révész, and Seoane-Sepúlveda [31]. First, it is convenient to have a formula to calculate0, 0 the0, norm 1 ‖ ⋅1, ‖훥 0.

T h e o r e m 1 7 . Let 푎 푏 푐 ∈ ℝ and 푃(푥 푦) = 푎푥 + 푏푦 + 푐푥푦. Then, | 푐 − 푎푏2 | 2 푏−푐 max {|푎| |푏| | (푎−푐+푏) |} if 푎 − 푐 + 푏 ≠ and < (푎−푐+푏) < ( 5 ) ‖푃‖, 훥, = { , 2 max{|푎| |푏|} 4 otherwise. 2 , , 4 0 0 2 1, Now we provide a parametrisation, of S 풫( 훥) and describe the geometry of B 풫( 훥). We use the notations S 훥 B and 훥 for short. 2 2

T h e o r e m 1 8 . If we define the mappings

푓+(푎 푏) = + √( − 푎)( − 푏) and , 2 2 1 1 푓−(푎 푏) = −푓+(−푎 −푏) = − − √( + 푎)( + 푏) for every (푎 푏) ∈ [− ] , and the set , , 2 2 1 1 , 2 퐹 = {(푎 푏 푐) ∈ ℝ ∶ (푎 푏) ∈ 휕[− ] and 푓−(푎 푏) ≤ 푐 ≤ 푓+(푎 푏)} , 1, 1 where 휕[− ] is the boundary of [−3 ] , then 2 , , , 1, 1 , , , 2 2 ( a ) S 훥 = graph(푓+|[− ] ) ∪ graph(푓−|[− ] ) ∪ 퐹. 1, 1 1, 1 2 2 ( b ) ext(B 훥) = {±( 푡 1,1− − √ ( + 푡)) 1,1±(푡 − − √ ( + 푡)) ∶ 푡 ∈ [− ]}.

You can find a picture of S 훥 in figure 2. 1, , 2 2 2 1 , , 1, 2 2 2 1 1, 1

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F i g u r e 2 : Unit sphere of 풫( 훥).

3 . 3 . The geom etry of 풫( 퐷(훼 훽))

First, recall that 풫( 퐷(훼 훽)) is the2 -dimensional space of the polynomials 푎푥 + 푏푦 + 푐푥푦 endowed with the norm 2 , 2 2 , ‖푃‖3 퐷(훼 훽) ≔ sup{|푃(x)| ∶ x ∈ 퐷(훼 훽)}.

It is a simple exercise to show that the spaces, 풫( 퐷(훼 훼 + 훽)) and 풫( 퐷( 훽)) are isometric. We write 퐷(훽) , instead of 퐷( 훽) for simplicity. Actually, the isometry2 is given by the2 matrix , 0, cos 훼 sin 훼 sin 훼 0, sin 훼 ( sin2 훼 cos2 훼 − 2 ) . 2 − sin2 훼 sin 2 훼 cos 2훼 2 This isometry allows us to restrict our attention to the study of the geometryof B . 2 2 2 퐷(훽) A moment’s thought reveals that, if 훽 ≥ π, then B 퐷(훽) = B 풫( ℓ ), where B 풫( ℓ ) stands for the closed unit ball of the space 풫( ℓ ) of -homogeneous polynomials on ℝ 2endowed2 with2 the2 sup norm over the unit disk. 2 2 The extreme2 points2 of B 풫( ℓ ) were described by Choi and2 Kim [10]. An alternative approach was provided by Muñoz-Fernández,2 Pellegrino, Seoane-Sepúlveda, and Weber [30]. 2 2 2 2

T h e o r e m 1 9 . Let 훽 ≥ π and define 푓(푎 푏) = √ + 푎푏 − |푎 + 푏| on [− ] . Then,

2 ( a ) ‖푃‖ = (|푎 + 푏| + √(푎 − 푏) + 푐 ), for all 푃 ∈ 풫( 퐷(훽)). 퐷(훽) , 2 1 1, 1 1 ( b ) S 풫( 퐷(훽)) = graph(푓) ∪ graph(−푓)2 . 2 2 2 2 ( c ) ext(B 풫( 퐷(훽))) = {±(푡 −푡 √ − 푡 ) ∶ 푡 ∈ [− ]} ∪ {±( )}.

2 2 The reader can find a graph of S in figure 3. , , 2 풫(1ℓ ) 1, 1 1, 1, 0 2 2 Let us give just another example in2 this section, taken from the work of Muñoz-Fernández et al. [30].

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F i g u r e 3 : S 풫( ℓ ). The extreme points of B 풫( ℓ ) are drawn with a thick line and dots.

2 2 2 2 2 2

T h e o r e m 2 0 . If we define the mappings

퐺 (푎 푏) = √( − 푎)( − 푏) and 1 , 2 1 1 퐺 (푎 푏) = −퐺 (−푎 −푏) = − √( + 푎)( + 푏) for every (푎 푏) ∈ [− ] , and the2 set 1 , , 2 1 1 , 2 퐹 = {(푎 푏 푐) ∈ ℝ ∶ (푎 푏) ∈ 휕[− ] and 퐺 (푎 푏) ≤ 푐 ≤ 퐺 (푎 푏)} , 1, 1 3 2 where 휕[− ] is the boundary of [− ] , then 2 1 , , , 1, 1 , , , 2 2 π ( a ) ‖푃‖ π = max {|푎| |푏| |푎 + 푏 + sign(푐)√(푎 − 푏) + 푐 |} for every 푃 ∈ 풫 ( 퐷 ( )). 퐷1,( 1) 1, 1 π 1 2 2 2 ( b ) S 퐷( ) 2= graph(퐺 ) ∪ graph(퐺 ) ∪ 퐹. , , 2 2 ( c ) ext(2B 퐷( π )) = {± (1 푡 − √ ( 2+ 푡)) ± (푡 − √ ( + 푡)) ∶ 푡 ∈ [− ]} ∪ {±( )}.

The reader can2 find a sketch of S π in figure 4. 1, , 2 2풫1( 퐷( )) , , 1, 2 2 1 1, 1 1, 1, 0 2 2

4 . Polynom ial inequalities

A number of polynomial inequalities can be tackled using the Krein-Milman approach described right after theorem 15. In this section we will introduce some problems of interest together with a sample of the type of results that can be achieved using the Krein-Milman approach.

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F i g u r e 4 : S 퐷( π ). The extreme points of B 퐷( π ) are drawn with a thick line and dots.

2 2

4 . 1 . The Bohnenblust-Hille inequality and related problem s

푚 푛 The ℓ푞 norm of the coefficients of polynomials in 풫( ) is given by

푞 푞 (∑ |푎훼| ) 1 if ≤ 푞 < +∞, |푃|푞 ≔ { |훼|=푚 max{|푎훼| ∶ |훼| = 푚} if 푞 = +∞, 1 푚 푛 for every 푃 ∈ 풫( ) with coefficients 푎훼. Observe that

푠 − 푞 | ⋅ |푞 ≤ | ⋅ |푠 ≤ 푑 | ⋅ |푞 1 1 for ≤ 푠 ≤ 푞, where 푑 is the dimension of 풫(푚푛). These norms appear in a number of problems of , 푚 푛 interest. They can also be used to estimate the difficult-to-calculate polynomial norm ofthespaces 풫( ℓ푝 ). 푚 푛 Let1 us denote the norm in 풫( ℓ푝 ) by ‖ ⋅ ‖푝. Since the norms | ⋅ |푞 and ‖ ⋅ ‖푝 are equivalent for all 푝 푞 ≥ , there exist 푘 and 퐾 depending on 푝 푞 푚 푛 such that , 1 푘‖푃‖ ≤ |푃| ≤ 퐾‖푃‖ , , , 푝 푞 푝 for all 푃 ∈ 풫(푚ℝ푛). The optimal values of the constants 푘 and 퐾 can be calculated in many situations using , the Krein-Milman approach. Indeed, as for the constant 퐾, the target function to which the Krein-Milman approach could be applied is

퐵‖⋅‖푝 ∋ 푃 ↦ |푃|푞.

TEMat monogr., 1 (2020) e-ISSN: 2660-6003 91 Geometry of polynomial spaces and polynomial inequalities

Here is where the geometry of 퐵‖⋅‖푝 can be used to optimise the target functions in the case when we have a description of its extreme points. The equivalence constants which we have just introduced are closely related to the famous polynomial Bohnenblust-Hille constants. Let us call 퐾푚 푛 푞 푝 the best (smallest) value of 퐾 in (4.1). The 푚-th polynomial Bohnenblust-Hille constant is nothing but an upper bound for 퐾 푚 . The reason why the specific 푚 푛 푚+ ∞ 푚 , , , 푚 choice 푞 = and 푝 = ∞ is of interest rests on the fact that, if 푞 ≥ 2 , then there exists a constant 푚+ , , 푚+1 , 퐷푚 푞 > depending2 only on 푚 and 푞 such that 2 1 1

( 6 ) , |푃| ≤ 퐷 ‖푃‖ 0 푞 푚 푞 ∞ 푚푛 ℕ , 푚 for all 푃 ∈ 풫( ) and every 푛 ∈ . Moreover, any constant, fitting in (6) for 푞 < 푚+ depends necessarily on 푛. This result was proved by Bohnenblust and Hille [7] in 1931. Observe that any2 plausible choice for 퐷푚 푞 in (6) must satisfy 퐷푚 푞 ≥ sup{퐾푚 푛 푞 ∞ ∶ 푛 ∈ ℕ}. The best (in the sense 1of smallest) possible choice 푚 for 퐷푚 푞 in (6) when 푞 = is called the polynomial Bohnenblust-Hille constant. It is interesting to , 푚+, , , , notice that there exists a considerable2 difference between the polynomial Bohnenblust-Hille constants for real and, complex polynomials.1 For this reason, the polynomial Bohnenblust-Hille constants are usually denoted by 퐷 푚. 푛 ∈ ℕ 퐷 (푛) > Moreover, if we, keep fixed, the best (smallest) 푚 in

|푃| 푚 ≤ 퐷푚(푛)‖푃‖∞ 푚+ 0 2 푚 푛 1 for all 푃 ∈ 풫( ), is denoted by 퐷 푚(푛). Observe that 퐷 푚(푛) = 퐾 푚 . The calculation of the , 푚 푛 푚+ ∞ Bohnenblust-Hille constants 퐷 푚 and 퐷 푚(푛) has motivated a large amount2 of papers, but their exact values are still unknown except for very, restricted choices of 푚, and 푛. The, , best1 , lower and upper estimates , , on 퐷 푚 and 퐷 푚(푛) known nowadays can be found in the literature [3, 8, 9, 12–14, 20, 25].

We present, below, a simple application of the Krein-Milman approach to calculate the value of 퐷ℝ( ) based on the following result by Choi, Kim, and Ki [11]. 2 T h e o r e m 2 1 B ℝ . The set ext( 풫( ℓ∞(ℝ))) of extreme points of the unit ball of 풫( ℓ∞( )) is given by

2 2 2 2 B √ ext( 풫( ℓ∞(ℝ))) = {±푥 ±푦 ±(푡푥 − 푡푦 ± 푡( − 푡)푥푦) ∶ 푡 ∈ [ / ]}.

2 2 2 2 2 2 T h e o r e m 2 2 . Let 푓 be the real-valued function given by , , 2 1 1 2, 1

√ 푓(푡) = [ 푡 + ( 푡( − 푡)) ] 3 . 4 4 4 3 3 We have that 퐷ℝ ( ) = 푓(푡 ) ≈ 1.837 373, where2 2 1

,2 0 2 푡 = ( √ + √ + √ − √ + ) ≈ 0.867 835. 3 3 0 1 Moreover, the following normalized2 107 polynomials9 129 are856 extreme72 129 for this16 problem: 36

푃 (푥 푦) = ± (푡 푥 − 푡 푦 ± √푡 ( − 푡 )푥푦) . 2 2 2 0 0 0 0 P r o o f . We just have to notice that,, due to the convexity of2 the ℓ1푝-norms and theorem 21, we have

B B 퐷ℝ ( ) = sup{|a| ∶ a ∈ 풫( ℓ∞ℝ)} = sup{|a| ∶ a ∈ ext( 풫( ℓ∞ℝ))} = sup 푓(푡). 푡∈[ / ] 4 2 2 4 2 2 ,2 3 3 The function 푓 is2 maximized using elementary calculus. The help of computer1 2,1 packages of symbolic calculus such as Matlab may be helpful to prove that 푓 attains its maximum in [ / ] at 푡 = 푡 , thus concluding the proof. ▪ 0 1 2, 2

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4 . 2 . Bernstein and M arkov type inequalities

Bernstein and Markov inequalities are estimates on the growth of the derivatives of polynomials. The famous Russian chemist D. Mendeleev (the author of the periodic table of elements) was among the pioneers that studied these types of estimates. In particular, he was interested in the following problem:

If 푝(푥) = 푎푥 + 푏푥 + 푐 with 푎 푏 푐 ∈ ℝ, 훼 훽 ∈ ℝ with 훼 < 훽, and we define ‖푝‖[훼 훽] ≔ max{|푝(푥)| ∶2 푥 ∈ [훼 훽]}, then what is the smallest possible constant 푀 (훼 훽) > so that ′ , |푝 (푥)| ≤ 푀 (훼 훽)‖푝‖[훼 훽] for every, , 푥 ∈ [훼 훽],and every quadratic polynomial 푝? 2 , , 0 2 , Considering an appropriate, change of variable,, namely 푥 → [훼 + 훽 + (훽 − 훼)푥]/ , it can be seen that 푀 (훼 훽) = /(훽 − 훼)푀 (− ) and, hence, we can restrict ourselves to (quadratic) polynomials on the standard interval [− ]. Mendeleev gave his own solution to the problem proving2 that 푀 (− ) = . 2 2 Mendeleev’s, 2 result was generalised1, 1 by A. A. Markov in 1889 for polynomials of arbitrary degree [26]. What 2 A. A. Markov proved1, was 1 that 1, 1 4

′ ( 7 ) ‖푃 ‖[− ] ≤ 푛 ‖푃‖[− ] 2 with equality for the 푛-th Chebyshev polynomial1,1 of the first1,1 kind, defined, for 푥 ∈ [− ], by 푇 (푥) = , 푛 cos(푛 arccos 푥). V. A. Markov [27] (brother of A. A. Markov) provided in 1892 a sharp estimate on the norm of the 푘-th derivative of a polynomial of arbitrary degree. A. A. Markov’s inequality (7) can1, be 1 improved in the inner points of [− ]. Let 푀푛(푥) be the optimal constant in

|푃′(푥)| ≤ 푀‖푃‖ for all 푃 ∈ 풫 (ℝ). 1, 1 [− ] 푛 푛 According to a classical result due to Bernstein1,1 [6], we have 푀푛(푥) ≤ in (− ). Both the uniform , √ −푥 Markov type estimates on the norm of the derivative and the pointwise estimates due to Bernstein have 2 been generalised in many different ways in the last century. One1 of the 1,most 1 popular generalisations is due to Harris in 2010 [19]. He proved that the old A. A. Markov constant 푛 is valid for polynomials on any real Banach space, that is, if 푃 is a polynomial of arbitrary degree 푛 on2 a real Banach space 퐸, then ‖퐷푃(푥)‖ ≤ 푛 ‖푃‖. Obviously, the constant 푛 is optimal in the general case too. Many Bernstein and

Markov type estimates2 can be obtained by applying2 the Krein-Milman approach. We present here a worked out example where the Markov and Bernstein optimal estimates are obtained for the space of trinomials 푚 푛 풫푚 푛 = {푎푥 + 푏푥 + 푐} endowed with the norm

푚 푛 푚 푛 , ‖푎푥 + 푏푥 + 푐‖푚 푛 = sup{|푎푥 + 푏푥 + 푐| ∶ 푥 ∈ [− ]}.

Observe that the polynomial 푎푥푚 + 푏푥푛 +, 푐 in 풫 can be identified with (푎 푏 푐) in ℝ . The geometry of 푚 푛 1, 1 the space 풫푚 푛 was studied by Muñoz-Fernández and Seoane-Sepúlveda [33], and the3 optimal value of , the Markov constant 푀푚 푛 and the Bernstein function 푀푚 푛(푥) where calculated, , by Muñoz-Fernández, Sarantopoulos,, and Seoane-Sepúlveda [32] when 푚 is odd and 푛 is even. We reproduce here the complete reasoning, based on the Krein-Milman, approach and the following, characterisation of the extreme points of B 푚 푛 (unit ball of 풫푚 푛) when 푚 is odd and 푛 is even.

T h e o r, e m 2 3 . If 푚 푛 ∈, ℕ are such that 푚 is odd, 푛 is even and 푚 > 푛, the extreme points of the unit ball of (ℝ ‖ ⋅ ‖푚 푛) are 3 , {±( − ) ±( − ) ±( − ) ±( )} . , , T h e o r e m 2 4 . Let 푚 푛 ∈ ℕ be such that 푚 is odd, 푛 is even and 푚 > 푛. Then, 0, 2, 1 , 1, 1, 1 , 1, 1, 1 , 0, 0, 1

푛− 푛 푚−푛 , 푛|푥| if ≤ |푥| ≤ ( 푚 ) ( 8 ) 푀푚 푛(푥) = { 1 푚푥푚− 1+ 푛|푥|푛− if ( 푛 ) 푚−푛 ≤ |푥| ≤ . 푚 1 , 2 0 , 1 1 P r o o f . If 푥 ∈ [− ], by definition we have that 1 푀 (푥) = sup |푝′(푥)|. 1, 1 푚 푛 푝∈B 푚 푛 , , TEMat monogr., 1 (2020) e-ISSN: 2660-6003 93 Geometry of polynomial spaces and polynomial inequalities

It suffices to work just with the extreme polynomials of 푀푚 푛, which are given in theorem 23. Notice that the contribution of ± to 푀푚 푛(푥) is irrelevant. Hence, it suffices to consider the polynomials , 푝 (푥) = ±( 푥푛 −, ) 푝 (푥) = ±(푥푚 + 푥푛 − ) and 푝 (푥) = ±(푥푚 − 푥푛 + ). 1 Therefore, 1 2 3 2 1 , 1 1 ′ ′ ′ 푀푚 푛(푥) = max{|푝 (푥)| |푝 (푥)| |푝 (푥)|} = max{ 푛|푥|푛− |푚푥푚− + 푛푥푛− | |푚푥푚− − 푛푥푛− |} , 1 2 3 푛− 푚− 푛− = max{ 푛|푥| , 1 푚푥 , 1+ 푛|푥| 1 } 1 1 푛− 푚−푛 = |푥| 2max{ 푛1, 푚|푥| 1 + 푛} 1,

12 , and since 푛 ≤ 푚|푥|푚−푛 + 푛 and ( 푛 ) 푚−푛 ≤ |푥|2 are, equivalent,, the result follows immediately. ▪ 푚 1

Corollary 25 . If 푚 푛 ∈ ℕ are such that 푚 is odd, 푛 is even and 푚 > 푛, then 2 푀 = 푀 (± ) = 푚 + 푛 , 푚 푛 푚 푛 and equality is attained for the polynomials, 푝(푥) =, ±(푥푚 + 푥푛 − ) and 푝(푥) = ±(푥푚 − 푥푛 + ). 1 ,

1 1 R e f e r e n c e s

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