# Convexity and the Euclidean Metric of Space-Time

## Abstract

**:**

## 1. Introduction

## 2. Background and Physical Interpretation of Some Concepts

#### 2.1. Norms on Linear Spaces

- Positive definiteness: $\parallel x\parallel =0$, if and only if $x=0$.
- Homogeneity: $\parallel kx\parallel =\left|k\right|\parallel x\parallel $, for all $x\in \mathcal{V}$, and all $k\in \mathbb{F}.$
- Triangle inequality: $\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel $

- ${c}_{0}$ is the space of all sequences $a=\left({a}_{n}\right),n\in \mathbb{N}$ converging to zero, with ${a}_{n}\in \mathbb{F}$, and the sup-norm$$\parallel a\parallel =\underset{n}{sup}\left|{a}_{n}\right|$$
- ${l}^{p},1\le p<\infty $, which is the space of all sequences $a=\left({a}_{n}\right),n\in \mathbb{N}$ with ${a}_{n}\in \mathbb{F}$ endowed with the norm$${\parallel a\parallel}_{p}={\left(\sum _{i=1}^{n}{\left|{a}_{n}\right|}^{p}\right)}^{\frac{1}{p}}<\infty $$
- ${l}^{\infty}$, the space of all bounded sequences $a=\left({a}_{n}\right),n\in \mathbb{N}$ endowed with the supremum norm$$\parallel a\parallel =\underset{n}{sup}\left|{a}_{n}\right|$$
- ${L}^{p}\left({\mathbb{R}}^{n}\right),1\le p<\infty $, the space of Lebesgue integrable functions $f:{\mathbb{R}}^{n}\to \mathbb{F}$ endowed with the norm$${\parallel f\parallel}_{p}={\left({\int}_{{\mathbb{R}}^{n}}{\left|f\right|}^{p}\right)}^{\frac{1}{p}}$$
- ${L}^{\infty}\left({\mathbb{R}}^{n}\right)$, the space of all essentially bounded $f:{\mathbb{R}}^{n}\to \mathbb{F}$, endowed with the norm$${\parallel f\parallel}_{\infty}=inf\{C:|f|<C\mathrm{almosteverywhere}\}$$

#### 2.2. Norm Equivalence

#### 2.3. The Operator Norm and the Banach-Mazur Distance

#### 2.4. Reflexive and Super-Reflexive Spaces

**Reflexive Spaces.**Let $\mathcal{X}$ be a Banach space and let ${B}_{\mathcal{X}}$ indicate its closed unit ball, namely

- the dual of reflexive space is reflexive.
- the closed subspaces of reflexive spaces are reflexive.
- the quotient spaces of reflexive spaces are reflexive, etc.

**Finite representability.**One may be able to demand a stronger property along the lines of reflexivity, from physically relevant Banach spaces, for the purposes of determining the space-time metric, First a definition: a Banach space $\mathcal{X}$ is finitely representable in a Banach space $\mathcal{Z}$ if for every $\u03f5>0$ and for every finite-dimensional subspace ${\mathcal{X}}_{0}\subset \mathcal{X}$ there is a subspace ${\mathcal{Z}}_{0}\subset \mathcal{Z}$ such that ${d}_{BM}({\mathcal{X}}_{0},{\mathcal{Z}}_{0})<1+\u03f5$. This essentially means that any finite-dimensional subspace of $\mathcal{X}$ can be represented, almost isometrically, in $\mathcal{Z}$. Equivalently, one controls the distortion of the embedding of every finite-dimensional subspace of $\mathcal{X}$ into $\mathcal{Z}$. From a physical viewpoint the above definition may be of interest, since it is at the confluence of two ideas: one has to do with the fact that based on quantum physics, or on the statistical interpretation of theories of many degrees of freedom, one may have to reconsider or even dispense with the concept of strict, “point-wise”, equality. Instead one should think much more along the lines of probabilistic equivalence, something that of course needs further qualifications. From such a perspective though, an approximate rather than strict, demand for isometry such as required in the definition of finite representability of a finite dimensional linear space is not unreasonable. The second idea relies on the fact that in any physical measurement we have a finite number of pieces of data on which to rely. As a result, the infinite dimensional spaces are excellent mathematical models, but from a very pragmatic perspective we see only their finite subspaces and then we mentally and technically extrapolate to the infinite dimensional counterparts. From this viewpoint, properties of finite dimensional vector spaces is the most of what someone can realistically expect to have to deal with in physical applications.

**Super-reflexive spaces.**A Banach space $\mathcal{X}$ is called super-reflexive if every Banach space which is finitely representable in $\mathcal{X}$ is reflexive. Equivalently, a Banach space $\mathcal{X}$ is super-reflexive if no non-reflexive Banach space $\mathcal{Y}$ is finitely representable in $\mathcal{X}$. Examples of super-reflexive spaces, pertinent to our discussion, are the Lebesgue spaces ${L}^{p}\left({\mathbb{R}}^{n}\right),1<p<n$. Super-reflexive spaces are reflexive, but the converse is not true. Super-reflexive spaces have numerous desirable properties, from a physical viewpoint, some of which will be encountered in the next Section as they are pertinent to convexity and smoothness properties. One property is that if a Banach space is isomorphic to a super-reflexive space then it is itself super-reflexive. Another useful property is is that a Banach space $\mathcal{X}$ is super-reflexive if and only if its dual ${\mathcal{X}}^{\prime}$ is super-reflexive. The super-reflexivity of Banach spaces is a property which allows the structure of infinite dimensional Banach spaces to be determined by the embedding properties of its finite-dimensional subspaces. Since ${c}_{0}$ and ${l}^{1}$ are not reflexive Banach spaces, they are not super-reflexive either. For completeness, we mention there are reflexive spaces are not necessarily super-reflexive: indeed consider a Banach space such that ${L}^{\infty}\left({\mathbb{R}}^{n}\right)$ is finitely representable in it; then it cannot be not super-reflexive. In closing, one would like to notice that super-reflexivity, very much like reflexivity, is a topological property: as it does not really depend on the specific norm with which the underlying linear space is endowed.

**Super-properties.**One can be more general at this point and talk about “super-properties”, a term that we will occasionally use in the sequel. These were defined by R.C. James in [27]. Here, we follow the excellent “pedestrian” exposition of [28]. Consider a property P that is valid on a Banach space $\mathcal{X}$. Consider two finite-dimensional subspaces $\mathcal{Y},\mathcal{Z}\subset \mathcal{X}$ and numbers ${n}_{P}\left(\mathcal{Y}\right),{n}_{P}\left(\mathcal{Z}\right)$ respectively such that

**The Radon-Nikodým property.**An additional property which is quite desirable at the technical level, and which is extensively used in Statistical Mechanics, where it is usually taken for granted, is the Radon-Nikodým property. It basically provides a way to make a transition between two different measures in a measure space. From a certain viewpoint, it can be seen as a generalization of the change of variables formula employed in multivariable calculus integration. For “practical purposes”, it states that one can use a function alongside the volume of a manifold as equivalent to any absolutely continuous measure. Such a density function is the micro-canonical distribution employed extensively in equilibrium Statistical Mechanics. This mathematical result extends to vector-valued measures as follows: consider a probability space ($\mathsf{\Omega},\mu $) with a σ- algebra of (Borel) sets Σ, $U\in \Sigma $ and a Banach space $\mathcal{X}$. Let the vector-valued measure $\nu :\Sigma \to \mathcal{X}$ be countably additive and of bounded variation. Then there is a (Bochner) integrable function $f:\mathsf{\Omega}\to \mathcal{X}$ such that

## 3. Convexity and Smoothness

#### 3.1. Why Convexity and Smoothness?

#### 3.2. A Modulus of Convexity

**Convex sets.**To be more precise and attempt to make the exposition somewhat self-contained, we present the following well-known definitions and statements. Consider $\mathcal{V}$ to be a vector space over $\mathbb{R}$ or $\mathbb{C}$. A subset $\mathcal{A}\subset \mathcal{V}$ is called (affinely) convex if

**Convex functions.**Convex functions can be considered as generalizations of convex sets. Let $\mathcal{A}\subset \mathcal{V}$ be a convex subset of the linear space $\mathcal{V}$ and let $f:\mathcal{A}\to \mathbb{R}$ be a function. The epigraph of f is defined to be the set

**A modulus of convexity.**A next logical step is to find a way to quantify the extent of convexity of a set or of a function. Naturally, determining such a “modulus of convexity” is not a unique process and it involves making certain choices. Simplicity and computability in, at least, simple cases are usually good guidelines, as far as physical applications are concerned. A relatively recent list of such moduli which is quite extensive, even if not necessarily comprehensive, can be found in [38].

**Uniformly convex spaces.**A Banach space ($\mathcal{X},\parallel \xb7\parallel $) is uniformly convex when it has non-zero modulus of convexity, namely ${\delta}_{\mathcal{X}}\left(\epsilon \right)>0,\epsilon \in (0,2]$ or equivalently when ${\epsilon}_{0}\left(\mathcal{X}\right)=0$. Geometrically, the idea of the definition is simple: uniformly convex spaces have a unit ball ${B}_{\mathcal{X}}$ whose boundary unit sphere ${S}_{\mathcal{X}}$ does not contain any (affine) line segments. Roughly speaking: the further away from containing an affine segment ${S}_{\mathcal{X}}$ is, the higher the modulus of convexity of $\mathcal{X}$ is. It should be noticed that according the D.P. Milman [43] - B.J. Pettis [44] theorem, uniformly convex spaces are reflexive. Actually one can see that uniformly convex spaces are actually super-reflexive. Hence, if one deems reflexivity or super-reflexivity to be a desirable, or pertinent, property in an argument about the Euclidean nature of the space-time metric, as was stated above, then confining their attention to uniformly convex Banach spaces will not miss this property.

**Modulus of convexity of Lebesgue spaces.**Explicitly calculating the modulus of convexity for specific Banach spaces has proved to be more difficult than one might have naively anticipated. This is one reason why so many different moduli of convexity have been defined over the decades, after Clarkson’s work [38]. For completeness, we mention that for $p=1$ and for $p=\infty $, ${\delta}_{{L}^{p}}=0$ as these two Banach spaces are not uniformly convex. The fact that the other Lebesgue spaces ${L}^{p}\left({\mathbb{R}}^{n}\right),1<p<\infty $ are uniformly convex was already known to [39]. For a simpler and more recent proof, see [47]. However, the explicit asymptotic form of their modulus of convexity was determined by [48] who relied on the inequalities bearing his name [48,49] to reach his result. Hanner proved that

**Modulus of convexity and equivalences.**It should be noted that the modulus of convexity ${\delta}_{\mathcal{X}}\left(\epsilon \right)$ is not necessarily itself a convex function of ε. What we know, for instance, is that for an infinite dimensional uniformly convex Banach space $\mathcal{X}$, we have that

#### 3.3. A Modulus of Smoothness and a Duality

**A modulus of smoothness.**The oldest and most studied modulus of smoothness is due to M.M. Day [40] and J. Lindenstrauss [58]. The modulus of smoothness of a normed space ($\mathcal{X},\parallel \xb7\parallel $) is a function ${\rho}_{\mathcal{X}}:[0,\infty )\to \mathbb{R}$ which is defined by

**Uniformly smooth spaces.**The Banach space ($\mathcal{X},\parallel \xb7\parallel $) is called uniformly smooth if ${\rho}_{0}\left(\mathcal{X}\right)=0$. One sees immediately that uniform smoothness is a point-wise property and is essentially 2-dimensional, as is the case for uniform convexity. In a pictorial sense, this modulus of smoothness captures the fact that the unit sphere ${S}_{\mathcal{X}}$ of the Banach space $\mathcal{X}$ is smooth, i.e., that it has not corners. More issue on this issue is discussed below.

**Modulus of smoothness of Lebesgue spaces.**Unlike the modulus of convexity, the modulus of smoothness is a convex function, essentially by definition. The asymptotic behavior of the modulus of smoothness of the Lebesgue spaces ${L}^{p}\left({\mathbb{R}}^{n}\right)$ were also calculated in [48]. O. Hanner found that

**A duality and the Legendre-Fenchel transforms.**One observes form the above that Clarkson’s modulus of convexity and the Day-Lindenstrauss modulus of smoothness appear to behave very much like dual concepts. The fact is that this suspected duality is true. More precisely, [58] proved that a Banach space $\mathcal{X}$ is uniformly convex if and only if its dual ${\mathcal{X}}^{\prime}$ is uniformly smooth. The exact relation between the corresponding moduli is given by the Legendre-Fenchel transform

#### 3.4. Smoothness, Derivatives and Equivalences

- $\mathcal{X}$ is super-reflexive.
- $\mathcal{X}$ admits an equivalent, uniformly convex norm, whose modulus of convexity satisfies, for some $q\ge 2$,$${\delta}_{\mathcal{X}}\left(\epsilon \right)\ge {c}_{1}{\epsilon}^{q}$$
- $\mathcal{X}$ admits an equivalent, uniformly smooth norm, whose modulus of smoothness satisfies, for some $1<p\le 2$$${\rho}_{\mathcal{X}}\left(t\right)\le {c}_{2}{t}^{p}$$

#### 3.5. Type, Co-Type and Moduli

**Rademacher functions.**The powers q and p in the lower bound of the modulus of convexity (52) and in the upper bound in the modulus of smoothness (54), respectively, have a nice geometric-probabilistic interpretation. To formulate it, we need to use the Rademacher functions which are defined as ${r}_{i}:[0,1]\to \pm 1$ by

**Type.**Suppose now that for any finite number n and any choice of vectors ${x}_{i},i=1,\dots ,n$ of $\mathcal{X}$ there is a constant ${C}_{p}>0$ such that

**Co-type.**With similar notation as for type, a Banach space $\mathcal{Y}$ has co-type q if there is a constant ${C}_{q}^{\prime}>0$ such that

**Type and co-type properties.**As an explicit example of type and co-type we know [30] that the Lebesgue spaces ${L}^{p}\left({\mathbb{R}}^{n}\right)$ have

- Type p and co-type 2, if $1\le p\le 2$
- Type 2 and co-type p, if $2\le p<\infty $

**Type, co-type and moduli.**The relation of the type and co-type of a Banach space with its moduli of convexity and smoothness is contained in the following theorem due to T. Figiel, G. Pisier [63]: Let $\mathcal{X}$ be a uniformly convex Banach space with modulus of convexity satisfying ${\delta}_{\mathcal{X}}\left(\epsilon \right)\ge C{\epsilon}^{q}$ for some $q\ge 2$. Then $\mathcal{X}$ has co-type q. Let $\mathcal{Y}$ be a uniformly smooth Banach space whose modulus of smoothness satisfies ${\rho}_{\mathcal{Y}}\left(t\right)\le c{t}^{p}$ for some $1<p\le 2$. Then $\mathcal{Y}$ has type p. Therefore the moduli of smoothness and convexity of a Banach space are bounded by the type and co-type of that Banach space, assuming that the latter exist. It may be worth noticing at this point the behavior of type and co-type under duality: It is known that when a Banach space $\mathcal{X}$ has type p, then its dual ${\mathcal{X}}^{\prime}$ has co-type q where p and q are harmonic conjugates of each other. However the converse is not true without one additional assumption. The accurate statement is that if a Banach space $\mathcal{X}$ has non-trivial type, and co-type q, then its dual ${\mathcal{X}}^{\prime}$ has type p, where p and q are harmonic conjugates of each other. For excellent expositions of the type and co-type of normed spaces, including proofs of all of the above statements, one may consult [30,31,62].

## 4. The Space-Time Metric from Variational Principles

**Functional integrals and variational principles.**Using extremal (more accurately: stationary) properties of functionals under infinitesimal variations subject to appropriate boundary conditions has been a fundamental aspect of Classical and Quantum Physics since the time of Maupertuis, D’Alembert and Lagrange at least [65], if not earlier. In particular, a large number of works in Quantum Physics have used and continue to use as starting point, especially for calculational purposes, the stationary phase or saddle point approximation which rely on the vanishing variation under small perturbations of a judiciously chosen functional (the “classical action” $\mathcal{S}$) [66], an approach that can be traced back to an original idea of P.A.M. Dirac [67]. In this path-integral approach, as is very well-known, one starts with the path-integral/canonical partition function as the primary object encoding the statistically significant properties of the system

**Other entropies and robustness.**Before proceeding, we would like to have a short digression. During the last three decades, there have been several functionals that have been proposed purporting to capture the collective/thermodynamic properties of systems of many degrees of freedom. One motivation for the formulation of such functionals, such as the “Tsallis entropy” [15] or the “κ-entropy” [16] is to determine the thermodynamic properties of systems with long-range interactions. From the Newtonian viewpoint, gravity clearly falls in this category, as well as Maxwell’s theory of electrodynamics etc. Assuming that such functionals may prove to be applicable to a path-integral formulation of aspects of semi-classical or even quantum gravity, the arguments of the present work will still hold without any major modifications. The minor modification needed in case such functionals are pertinent, is to use in (70) another convex function instead of the exponential one, something akin to the aptly named “q-exponential” [15], whose form will have to be determined. One could possibly use the maximum entropy principle subject to appropriate constraints, for such a purpose. A second minor, for our purposes, modification may be to substitute some other measure in the place of the often used Gaussian measure in (70). Beyond these points, we expect the above analysis to still be valid. Another point that will most likely change is the rate of convergence to the limit of the saddle-point approximation, which is intimately related to the probability of dealing with space-time geometries that may be non-Euclidean. This in the spirit of theories having a statistical interpretation, where even when the “classical” limit is known, it is the form of the “semi-classical” contributions/corrections that is used to distinguish between several competing models purporting to describe the same physical phenomenon.

**Generalized path integrals.**Going back to our argument, in the spirit of the path-integral, an often discussed but still unsolved question is whether one should extend (70) by considering additional contributions by summing over more “primitive”, than the metric, structures such over all topological, piecewise-linear, differentiable etc. structures. Most of the treatments to quantum gravity that we are aware of, address the issue of a possible sum in the right-hand-side of (70) over all topologies. Then the modification of (70) states that the partition function of quantum gravity should be

**A “kinematic” approach via smoothness.**Given the above difficulties, we have to resort to ad hoc decisions in order to proceed. To the extent of our knowledge, there has never been a variational principle of “maximum convexity” or a principle of “minimal smoothness” that would single out Hilbert spaces among all Banach spaces. The path-integral/partition function approach to quantum Physics can be interpreted as suggesting that all allowed possibilities in quantum evolution should be considered in calculating quantities of interest, each possibility however being assigned with a different weight factor. Following this viewpoint one can extend/stretch the domain of this interpretation to allow not only for a set of Riemannian metrics to contribute to the evolution of a gravitational system but also consider a broader class of possible metrics. To keep things close to the familiar territory of Riemannian/Lorentzian metrics we have used induced metrics on space-time locally induced by ${L}^{p}\left({\mathbb{R}}^{n}\right)$, as was mentioned before. The familiar picture of space-time appears then as the classical limit of a theory of quantum gravity, and so are its associated properties like smoothness, etc.

**Convexity and predictability.**A somewhat complementary argument for Hilbert spaces and the induced Euclidean metric form on space-time, can be made based on convexity and predictability. As stated in the previous sections, the Hilbert spaces $\mathcal{H}={L}^{2}\left({\mathbb{R}}^{n}\right)$ are the most convex among all ${L}^{p}\left({\mathbb{R}}^{n}\right)$. Contrast the behavior of the norm/metric of $\mathcal{H}$ to those of the family of ${L}^{p}\left({\mathbb{R}}^{n}\right)$ that are the least convex. These are ${L}^{1}\left({\mathbb{R}}^{n}\right)$ and ${L}^{\infty}\left({\mathbb{R}}^{n}\right)$ which are neither uniformly convex nor uniformly smooth, nor are they reflexive, so they have been largely ignored in most part of this work. Nevertheless, use of these two spaces can help make this argument more transparent.

**The space-time metric from Hilbert spaces.**Going from ${L}^{2}\left({\mathbb{R}}^{n}\right)$ to the metric of space-time itself is quite straightforward, in principle. Consider as the linear spaces of interest to be appropriate $A\subset {\mathbb{R}}^{n}$. For the case of point particles this will be the tangent to the particles’s space-time evolution trajectory. We can then confine ourselves to the analysis of a subspace of ${L}^{2}\left({\mathbb{R}}^{n}\right)$ which is comprised of the characteristic functions ${\chi}_{A}$ of such subspaces. Then the quadratic metric on ${L}^{2}\left({\mathbb{R}}^{n}\right)$ gets induced on such A which acquires itself a quadratic metric. Use the equivalence principle and “patch together” such A endowed with their Euclidean metrics to form the space-time of interest. This is a kinematic construction. The dynamics is provided by Einstein’s equations, after a Wick rotation back to indefinite signature metrics. This transition between metrics of different signatures may involve several subtleties which may have to be addressed at that stage, but this is outside the scope of the present work.

## 5. Discussion and Outlook

## Acknowledgments

## Conflicts of Interest

## References

- Pettini, M. Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics. In Interdisciplinary Applied Mathematics; Springer: New York, NY, USA, 2007; Volume 33. [Google Scholar]
- Kriegl, A.; Michor, P.W. The Convenient Setting of Global Analysis. In Mathematical Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 1997; Volume 53. [Google Scholar]
- Corichi, A.; Rubalcava-García, I.; Vukašinac, T. Actions, topological terms and boundaries in first order gravity: A review. Int. J. Mod. Phys. D
**2016**, 25, 1630011. [Google Scholar] [CrossRef] - Kar, A.; Rajeev, S.G. A Non-Riemannian Metric on Space-Time Emergent from Scalar Quantum Field Theory. Phys. Rev. D
**2012**, 86, 065022. [Google Scholar] [CrossRef] - Sorkin, R.D. Is the spacetime metric Euclidean rather than Lorentzian? In Recent Research in Quantum Gravity; Dasgupta, A., Ed.; Nova Science Publishers Inc.: New York, NY, USA, 2013. [Google Scholar]
- Bojowald, M. Canonical Relativity and the Dimensionality of the World. In Relativity and the Dimensionality of the World; Petkov, V., Ed.; Springer: New York, NY, USA, 2007; pp. 137–152. [Google Scholar]
- Nawarajan, D.; Visser, M. Global properties of physically interesting Lorentzian spacetimes. arXiv, 2016; arXiv:1601.03355. [Google Scholar]
- Gibbons, G.W.; Ishibashi, A. Convex Functions and Spacetime Geometry. Class. Quant. Grav.
**2001**, 18, 4607–4628. [Google Scholar] [CrossRef] - Rudin, W. Functional Analysis, 2nd ed.; McGraw-Hill Inc.: New York, NY, USA, 1991. [Google Scholar]
- Lieb, E.H.; Loss, M. Analysis, 2nd ed.; American Mathematical Society: Providence, RI, USA, 2001. [Google Scholar]
- Hájek, P.; Santalucía, V.M.; Vanderwerff, J.; Zizler, V. Biorthogonal Systems in Banach Spaces; Springer: New York, NY, USA, 2008. [Google Scholar]
- Simon, B. The classical moment problem as a self-adjoint finite difference operator. Adv. Math.
**1998**, 137, 82–203. [Google Scholar] [CrossRef] - Weinstein, A. The local structure of Poisson manifolds. J. Diff. Geom.
**1983**, 18, 523–557. [Google Scholar] - Klauder, J. Enhanced quantum procedures that resolve difficult problems. Rev. Math. Phys.
**2015**, 27, 1530002. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Kaniadakis, G. Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. Entropy
**2013**, 15, 3983–4010. [Google Scholar] [CrossRef] - Roe, J. Lectures on Coarse Geometry; American Mathematical Society: Providence, RI, USA, 2003. [Google Scholar]
- Bridson, M.R.; Haefliger, A. Metric Spaces of Non-Positive Curvature; Springer: Berlin, Germany, 1999. [Google Scholar]
- Gromov, M. Hyperbolic Groups. In Essays in Group Theory; Gesten, S.M., Ed.; Springer: New York, NY, USA, 1987; pp. 75–264. [Google Scholar]
- Katok, A.; Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Heinonen, J. Nonsmooth calculus. Bull. Amer. Math. Soc.
**2007**, 44, 163–232. [Google Scholar] [CrossRef] - Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal.
**1999**, 9, 428–517. [Google Scholar] [CrossRef] - Keith, S. A differentiable structure for metric measure spaces. Adv. Math.
**2004**, 183, 271–315. [Google Scholar] [CrossRef] - Cheeger, J.; Kleiner, B. Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal.
**2009**, 19, 1017–1028. [Google Scholar] [CrossRef] - Bate, D. Structure of measures in Lipschitz differentiability spaces. J. Amer. Math. Soc.
**2015**, 28, 412–482. [Google Scholar] [CrossRef] - Cheeger, J.; Kleiner, B.; Schioppa, A. Infinitesimal structure of differentiability spaces, and metric differentiation. arXiv, 2016; arXiv:1503.07348. [Google Scholar]
- James, R.C. Some self-dual properties of normed linear spaces. In Symposium on Infinite Dimensional Topology, 1967; Princeton University Press: Princeton, NJ, USA, 1972; pp. 159–175. [Google Scholar]
- Maurey, B. Type, Cotype and K-convexity. In Handbook of the Geometry of Banach Spaces; Johnson, W.B., Lindenstrauss, J., Eds.; Elsevier Science B.V.: Amsterdam, The Netherlands, 2003; pp. 1299–1332. [Google Scholar]
- Amir, D. Characterizations of Inner Product Spaces; Springer: Basel, Switzerland, 1986. [Google Scholar]
- Lindenstrauss, J.; Tzafriri, L. Classical Banach Spaces I and II; Reprint of the 1977 and 1979 Editions; Springer: Berlin, Germany, 1996. [Google Scholar]
- Benyamini, Y.; Lindenstrauss, J. Geometric Nonlinear Functional Analysis; American Mathematical Society: Providence, RI, USA, 2000. [Google Scholar]
- Rockafellar, R.T. Convex Analysis; Princeton University Press: Princeton, NJ, USA, 1970. [Google Scholar]
- Verpoort, S. The Geometry of the Second Fundamental form: Curvature Properties and Variational Aspects. Ph.D. Thesis, Katholieke Universiteit Leuven, Leuven, Belgium, May 2008. [Google Scholar]
- Bishop, R. Infinitesimal convexity implies local convexity. Ind. Univ. Math. J.
**1974/1975**, 24, 168–172. [Google Scholar] [CrossRef] - Chruściel, P.T.; Galloway, G.J.; Pollack, D. Mathematical general relativity: A sampler. Bull. Amer. Math. Soc.
**2010**, 47, 567–638. [Google Scholar] [CrossRef] - Parikh, M. Two Roads to the Null Energy Condition. Int. J. Mod. Phys. D
**2015**, 24, 1544030. [Google Scholar] [CrossRef] - Cordero-Erausquin, D.; McCann, R.J.; Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math.
**2001**, 146, 219–257. [Google Scholar] [CrossRef] - Fuster, E.L. Moduli and Constants: ... what a show! Available online: http://www.uv.es/llorens/Documento.pdf (accessed on 5 November 2016).
- Clarkson, J.A. Uniformly convex spaces. Trans. Amer. Math. Soc.
**1936**, 40, 394–414. [Google Scholar] [CrossRef] - Day, M.M. Uniform convexity in factor and conjugate spaces. Ann. Math.
**1944**, 45, 375–385. [Google Scholar] [CrossRef] - Gromov, M. Sign and geometric meaning of curvature. Rend. Mat. Sem. Mat. Fis. Milano Milan J. Math.
**1991**, 61, 9–123. [Google Scholar] [CrossRef] - Goebel, K. Convexity of balls and fixed point theorems for mapping with nonexpansive maps. Comp. Math.
**1970**, 22, 269–274. [Google Scholar] - Milman, D.P. On some criteria for the regularity of spaces of the type (B). Dokl. Acad. Nauk. SSSR
**1938**, 20, 243–246. [Google Scholar] - Pettis, B.J. A proof that every uniformly convex space is reflexive. Duke Math. J.
**1939**, 5, 249–253. [Google Scholar] [CrossRef] - Enflo, P. Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math.
**1972**, 13, 281–288. [Google Scholar] [CrossRef] - James, R.C. Super-reflexive Banach spaces. Can. J. Math.
**1972**, 24, 896–904. [Google Scholar] [CrossRef] - Hanche-Olsen, H. On the uniform convexity of L
^{p}. Proc. Amer. Math. Soc.**2006**, 134, 2359–2362. [Google Scholar] [CrossRef] - Hanner, O. On the uniform convexity of L
^{p}and l^{p}. Arkiv. Mat.**1956**, 3, 239–244. [Google Scholar] [CrossRef] - Ball, K.; Carlen, E.A.; Lieb, E.H. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math.
**1994**, 115, 463–482. [Google Scholar] [CrossRef] - Nordlander, G. The modulus of convexity in normed linear spaces. Arkiv. Mat.
**1960**, 4, 15–17. [Google Scholar] [CrossRef] - Day, M.M. Some characterizations of inner product spaces. Trans. Amer. Math. Soc.
**1947**, 62, 320–337. [Google Scholar] [CrossRef] - Schoenberg, I.J. A remark on M.M. Day’s characterization of inner product spaces and a conjecture of L.M. Blumenthal. Proc. Amer. Math. Soc.
**1952**, 3, 961–964. [Google Scholar] - Figiel, T. On the moduli of convexity and smoothness. Stud. Math.
**1976**, 56, 121–155. [Google Scholar] - Grigorchuk, R.; Pak, I. Groups of intermediate growth, an introduction. L’Enseign. Math.
**2008**, 54, 251–272. [Google Scholar] - Kalogeropoulos, N. Tsallis entropy composition and the Heisenberg group. Int. J. Geom. Methods Mod. Phys.
**2013**, 10, 1350032. [Google Scholar] [CrossRef] - Kalogeropoulos, N. Groups, non-additive entropy and phase transitions. Int. J. Mod. Phys. B
**2014**, 28, 1450162. [Google Scholar] [CrossRef] - Scott, P. The geometries of 3-manifolds. Bull. London Math. Soc.
**1983**, 15, 401–487. [Google Scholar] [CrossRef] - Lindenstrauss, J. On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J.
**1963**, 10, 241–252. [Google Scholar] [CrossRef] - Asplund, E. Averaged norms. Israel J. Math.
**1967**, 5, 227–233. [Google Scholar] [CrossRef] - Kwapień, S. Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math.
**1972**, 44, 583–595. [Google Scholar] - Kahane, J.P. Some Random Series of Functions, 2nd ed.; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Milman, V.D.; Schechtman, G. Asymptotic Theory of Finite Dimensional Normed Spaces; Springer: Berlin, Germany, 1986. [Google Scholar]
- Figiel, T.; Pisier, G. Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses. C. R. Acad. Sci. Paris Ser. A
**1974**, 279, 611–614. [Google Scholar] - Artstein-Avidan, S.; Milman, V. The concept of duality in convex analysis, and the characterization of the Legendre transform. Ann. Math.
**2009**, 169, 661–674. [Google Scholar] [CrossRef] - Lanczos, C. The Variational Principles of Mechanics, 4th ed.; University Toronto Press: Toronto, ON, Canada, 1970. [Google Scholar]
- Zinn-Justin, J. Quantum Field Theory and Critical Phenomena, 3rd ed.; Oxford University Press: Oxford, UK, 1996. [Google Scholar]
- Dirac, P.A.M. The Lagrangian in Quantum Mechanics. In Physikalische Zeitschrift der Sowjetunion; Charkow, Technischer staatsverlag: Charkow, Ukraine, 1933; pp. 64–72. [Google Scholar]
- Ambjørn, J.; Görlich, A.; Jurkiewicz, J.; Loll, R. CDT and the Search for a Theory of Quantum Gravity. arXiv, 2013; arXiv:1305.6680. [Google Scholar]
- Veltman, M.J.G. Quantum Theory of Gravitation. In Methods in Field Theory, Proceedings of the Les Houches Summer School of Theoretical Physics, Session 28, 28 July–6 September 1975; Balian, R., Zinn-Justin, J., Eds.; North Holland Publisher: Amsterdam, The Netherlands, 1981. [Google Scholar]
- Villani, C. Synthetic Theory of Ricci Curvature Bounds, Takagi Lectures 2015. Jpn. J. Math.
**2016**, 11, 219–263. [Google Scholar] [CrossRef] - Burago, D.; Burago, Y.; Ivanov, S. A Course in Metric Geometry; American Mathematical Society: Providence, RI, USA, 2001. [Google Scholar]
- Laskin, N. Fractional Quantum Mechanics. Phys. Rev. E
**2000**, 62, 3135–3145. [Google Scholar] [CrossRef] - Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media; Springer: New York, NY, USA, 2011. [Google Scholar]
- Calgagni, G. Geometry and field theory in multi-fractional spacetime. J. High Energy Phys.
**2012**, 2012, 65. [Google Scholar] [CrossRef] - Ashtekar, A.; Lewandowski, J. Background Independent Quantum Gravity: A Status Report. Class. Quant. Grav.
**2004**, 21, R53. [Google Scholar] [CrossRef] - Beem, J.K.; Ehrlich, P.E.; Easley, K.L. Global Lorentzian Geometry, 2nd ed.; Marcel Dekker Inc.: New York, NY, USA, 1996. [Google Scholar]
- Minguzzi, E.; Sánchez, M. The causal hierarchy of spacetimes. In Recent Developments in Pseudo-Riemannian Geometry; Alekseevsky, D.V., Baum, H., Eds.; ESI Lectures in Mathematics and Physics; European Mathematical Society: Zürich, Switzerland, 2008; pp. 299–358. [Google Scholar]

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Kalogeropoulos, N. Convexity and the Euclidean Metric of Space-Time. *Universe* **2017**, *3*, 8.
https://doi.org/10.3390/universe3010008

**AMA Style**

Kalogeropoulos N. Convexity and the Euclidean Metric of Space-Time. *Universe*. 2017; 3(1):8.
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Kalogeropoulos, Nikolaos. 2017. "Convexity and the Euclidean Metric of Space-Time" *Universe* 3, no. 1: 8.
https://doi.org/10.3390/universe3010008