# Ricci Curvature, Isoperimetry and a Non-additive Entropy

## Abstract

**:**

## 1. Introduction

_{i}. Then the non-additive entropy ${\mathcal{S}}_{q}$ that we will be interested us in this work, is defined by

_{B}stands for the Boltzmann constant which will be set k

_{B}= 1 almost everywhere in the sequel. The naive extension to a continuous set of outcomes, is characterised by the probability density function ρ : Ω → R, and is assumed to be absolutely continuous everywhere on Ω with respect to the Lebesgue measure (volume) dvol

_{Ω}is

**R**, although a recent work has suggested [7] the possibility of q ∈

**C**. We called the definition of ${\mathcal{S}}_{q}$ for continuous sets of outcomes (2) as “naive” since there has been some recent controversy about the validity of (2) [8–15] and some related skepticism on whether it is possible to extend ${\mathcal{S}}_{q}$ to continuous sets of outcomes. We consider this criticism to be valid and the general controversy not yet settled. Nevertheless, in the absence of a viable alternative or a consensus, we will use (2) in the sequel as the version of ${\mathcal{S}}_{q}$ for a continuous sets of outcomes, having the above possible caveat in mind. It is reassuring to notice that for q → 1 one recovers the Boltzmann/Gibbs/Shannon (BGS) entropic functional

## 2. About the Ricci Curvature and Its Generalizations

#### 2.1. Geometry in Mechanics

_{i}, i =1, …, n. Let its configuration space M be parametrized by the local coordinates {x

^{i}, x

^{j}}, i, j = 1, …, n and let its Lagrangian be

_{ij}stands for a positive definite quadratic form (mass matrix) which can be used as a Riemannian metric on M. In the above case (6), when all masses of particles are taken as equal to each other

_{i}= δ$\mathcal{L}$/δẋ

^{i}, its Hamiltonian

^{i}

_{jk}are given, as usual, in terms of the metric tensor components g

_{ij}by

_{i}= ∂/∂x

^{i}and the Einstein summation convention is assumed over repeated indices in (9)–(12) and henceforth, unless otherwise stated.

_{ij}metric, is the Jacobi metric

_{0}to describe the evolution of the system, but rather a set of them, all of which are initially close to c

_{0}. The behavior of nearby geodesics is encoded, in the linear approximation, via the Jacobi fields with coordinate components J

^{i}, i = 1, …, n. In their dynamical evolution, the infinitesimal separation between nearby geodesic is determined by the by the geodesic deviation (Jacobi) equation. This is a linearization of a variation of the equation of geodesics and, expressed in the local coordinates we are using, it is

^{i}

_{jkl}are the components of the Riemann tensor in the coordinate basis of TM.

#### 2.2. Rudiments of Riemannian Curvature

^{i}

_{jkl}is a fundamental object in Riemannian geometry. It should be noted that despite a century and a half of intense exploration, many of its properties still remain unknown [17]. A way to motivate its introduction is by searching for a local quantity that allows us to determine the distance between two points in a Riemannian manifold. Such a calculation is practically intractable, as can be seen by looking at the geodesic equation (11) in conjunction with (12). The Riemann tensor is a quantity that allows us to infinitesimally address such a question. This viewpoint, among several others, can be seen in [17].

^{i}, i = 1, …, n points from c

_{0}toward nearby geodesics, as noted above, but may also point along c

_{0}itself. The definition of the Riemann tensor is far more elegant and brings forth its linear and differential nature when expressed in terms of the Levi-Civita connection ∇ (the unique symmetric connection preserving the metric) as a multi-linear map R : TM × TM × TM → TM

^{i}can be chosen to be perpendicular to c

_{0}. We see from (16), that the behavior of nearby geodesics is controlled by the Riemann curvature tensor (17). Consideration of initial conditions near c

_{0}amounts to averaging in the n-1 perpendicular directions to c

_{0}, as expressed by J

^{i}, i = 2, …, n, by using a suitably chosen measure. The simplest case is to choose a uniform measure, meaning a measure which is a constant multiple of the Lebesgue measure (“Riemannian volume”) dvol. Then evolution of a set of initial conditions along c

_{0}is controlled by the average of the Riemann tensor in the 2-planes spanned by the tangent to c

_{0}and one of the perpendicular directions to c

_{0}, an average that results in the Ricci tensor along c

_{0}. Therefore, the Ricci tensor is given by the following, essentially unique, contraction of the Riemann tensor

_{i}}, i = 1, …, n. Let e

_{1}be the tangent to c. The sectional curvature in a 2-plane of TM at P ∈ M spanned by e

_{p}, e

_{q}, p ≠= q, p, q = 1, …, n is defined by

_{2}

_{,n}(M) of 2-dimensional planes of TM. So, the sectional curvature is, at its core, an essentially 2-dimensional quantity. Its geometric meaning becomes evident through the following two theorems which are true in 2-dimensions. Let P be a point of the Riemannian manifold M and let r be the radius of a geodesic circle around P. Let c(r) be its circumference and A(r) be the area of the geodesic disk whose boundary has length c(r). Then [18] Bertrand-Puiseux (1848)

#### 2.3. About the Ricci Curvature

_{1}, which as was stated above is assumed to be tangent to the geodesic c passing by P, is a symmetric bilinear form related to the Ricci tensor by

_{1}at P as was previously mentioned. This averaging is also explicit in the contraction in the indices of the Riemann tensor resulting in the Ricci tensor (18). Hence the definition of the Ricci tensor or Ricci curvature uses explicitly two distinct facets of Riemannian manifolds: their metric (and connection which is uniquely defined through the metric) and a measure (in this case the volume). These two concepts are uniquely inter-related in Riemannian manifolds exactly because such manifolds M are locally isometric, to first order, to the Euclidean space R

^{n}[17]. This can be most easily seen via the expansion of the metric in geodesic normal coordinates around P ∈ M taken as the origin of the normal coordinate system

- For surjective, distance-decreasing maps f : M
_{1}→ M_{2}, the volume obeys vol M_{1}≤ vol M_{2}. - The volume of the unit cube [0, 1]
^{n}in R^{n}is normalized so that vol ([0, 1]^{n}) = 1.

_{P}(r) be the geodesic ball of radius r centered at P and B

_{k}(r) be the ball of radius r in the space form of constant sectional curvature k and dimension n. Then the function

_{μν}g

^{μ}Einstein’s equations

_{μν}in terms of the stress energy tensor T

_{μν}. Here c stands for the speed of light and G is the universal gravitational constant. Roughly speaking, in areas where there is a lot of matter, T

_{μν}will result in strong gravity, so the bound in the Bishop-Gromov inequality will be high. Consider a homogeneous and isotropic space-time, as in the case of the Friedmann-Robertson-Walker (FRW) cosmology, as a comparison space. Any space-time that has more attractive matter than that, will make the geodesics converge faster, as seen also in (21), hence the volume of a ball centered along one such member of the geodesic congruence will contract faster than in the model (FRW) space. The above analogy relies in the fact that the results we use in the Riemannian signature carry over to the Lorentzian signature case, something that is known, even though non-trivial to prove [27].

_{1}of the Laplacian ∆ in terms of a lower bound on the Ricci curvature. As a reminder, the Laplacian ∆ on functions f : M → R on a Riemannian manifold (M, g) is defined to be the trace of the Hessian

^{i}, i = 1, …, n} in terms of the components of the metric g

_{ij}

_{1}for Physics is substantial: in the particular context of Lagrangian field theories, it expresses the lowest excitation of the mass spectrum, or alternatively and depending on the interpretation, the mass of the lightest particle in the particle spectrum. Since it is not practically feasible to explicitly calculate λ

_{1}even for the simplest manifolds, with scant few exceptions, providing bounds to it is the best that someone can hope for. Lichnerowicz’s theorem states that for (M, g) a compact Riemannian manifold without boundary with

_{1}satisfies

_{1}of the sphere S

^{n}endowed with the round metric, having constant curvature k/(n − 1). A result by Obata [29] states that the equality is attained if and only if (M, g) is actually isometric to such a round sphere. This result can be interpreted as stating that knowing the mass of the lowest excitation, alongside the Ricci curvature lower bound (32), uniquely determines the whole spectrum of excitations in this space(-time). It actually determines much more than that: it completely determines the geometry and the topology of such a space(-time). In General Relativity, or diffeomorphism-invariant theories, the Ricci curvature bound (31) usually results from imposing a strong energy condition [27], namely a lower bound to

#### 2.4. Generalized Ricci Curvature

_{E}gives rise to the micro-canonical measure. Hence it is of interest to determine a generalisation of the Ricci tensor that is defined with respect to the measure

_{ij}. Symmetric tensors constructed from f and involving two derivatives are (∂

_{i}f)(∂

_{j}f) as well as the Hessian

^{i}, i = 1, …, n the Hessian can be written as

_{N})

_{ij}by

_{i}f)(∂

_{j}f) can also be included with an arbitrary undetermined coefficient N ∈ [1, ∞) and the exploration of some of its consequences was provided first by Qian [32]. Therefore, a more general definition of R

_{N}is provided by

_{N→∞}of (40) reduces to (39). Putting all these elements together, one can state that the central differential quantity for characterising the non-ergodic behaviour of systems conjecturally described macroscopically by ${\mathcal{S}}_{q}$ is the generalized-/ N- / Bakry-Émery- Ricci tensor defined by

_{N}is, in reality, a one-parameter family of tensors giving non-trivial results when N ∈ [n, ∞].

_{N})

_{ij}= λg

_{ij}resulting from the definition of (41) for N < ∞. In the Physics literature, the generalised Ricci curvature has been recently discussed in explorations of scalar-tensor (e.g., dilaton, Brans-Dicke etc.) theories of gravity [46–48].

_{N})

_{ij}starts becoming clearer when one tries to check on whether, or under what conditions, standard results of Riemannian geometry can be extended to smooth metric measure spaces [34,35,49,50]. A theorem in the spirit of the Bishop-Gromov volume comparison goes as follows: Consider a compact, smooth metric measure space (M, g, μ) where following (35),

_{N})

_{ij}≥ kg

_{ij}, k ≥ 0 for some N ∈ [1, ∞). Then for all 0 < r ≤ R and P ∈ M one has

_{1}, φ

_{2}∈ L

^{2}(M, μ), one has

_{N})

_{ij}≥ kg

_{ij}, with k > 0, then

_{1}≥ k which is independent of N. This has important implications in the sequel as the N = ∞ case will turn out to be related to ${\mathcal{S}}_{BGS}$ when a finite N is related to ${\mathcal{S}}_{q}$.

## 3. Ricci Curvature via Optimal Transport

_{N}. For a comprehensive treatment of these topics, one can start by consulting [35].

#### 3.1. Otto’s View: the Porous Medium Equation and the Geometry of Space of Probability Distributions

^{n}

^{n}which explicitly depends on time t ∈ [0, ∞). In this equation $m\ge \frac{n-1}{n}$ and $m>\frac{n}{n+2}$ for reasons that will be stated later. The question that arose was how to interpret the porous medium equation as a gradient flow equation. As a reminder, a gradient flow [40–42,52] of sufficient generality for our purposes, consists of a Riemannian manifold (M, g) and an “energy” functional E[ρ] on (M, g) obeying the autonomous differential equation

^{n}). Its tangent space at ρ ∈ M, T

_{ρ}M, is

_{ρ}M can also be seen as

_{B}. To determine the scaling behaviour of a solution in the asymptotic regime close to ρ

_{B}, we re-express it as

^{2}being the Euclidean norm of {x

^{i}, i = 1, … n} ∈ R

^{n}. Otto proved [51] that $\tilde{E}$ is uniformly strictly convex on (M, g) since it satisfies

_{B}of E and V. Third, we observe in (64) that the rate of convergence of ρ to ρ

_{B}is polynomial (power-law) in terms of t. This, conjecturally, is one of the important properties of systems described by ${\mathcal{S}}_{q}$: their power law rate of convergence to their long-time asymptotic limits [6]. It is worth mentioning that this rate of convergence is believed to also have the form of a q-exponential

#### 3.2. Optimal Transportation and Wasserstein Spaces

^{n}, it can clearly be formulated for general complete separable metric (Polish) spaces endowed with Borel probably measures. It may be worth mentioning right away, that the general problem remains unsolved even for simple cost functions in R

^{n}, although substantial progress has been made in special cases, especially during the last two decades.

^{n}× R

^{n}.

^{+}, μ

^{−}on R

^{n}, or in more general Polish measure spaces mentioned above, such that

^{n}→ R

^{n}pushing forward μ

^{+}to μ

^{−}namely

^{n}→ R with A

_{1}, A

_{2}⊂ R

^{n}indicating the support of μ

^{+}, μ

^{−}respectively. Let the set of such functions be indicated by F. Given a cost function c : R

^{n}×R

^{n}→ [0, ∞) consider the total cost functional

- The problem is highly non-linear. To see this more concretely, let’s assume that both μ
^{+}, μ^{−}are absolutely continuous with respect to the volume element of R^{n}with corresponding Radon-Nikodym densities ρ^{+}, ρ^{−}. Then it turns out that the push forward condition (68) translates into the Monge-Ampére, non-linear, equation$${\rho}^{+}(x)={\rho}^{-}(f(x))\phantom{\rule{0.2em}{0ex}}\mathrm{det}(\nabla f)$$ - Such a solution may not exist: consider for instance μ
^{+}to be the Dirac delta function but not μ^{−}.

_{o}that we seek cannot “split” the measure dμ

^{+}. Heuristically, $f\in \mathcal{F}$cannot move a part of dμ

^{+}in a location of R

^{n}and another part in some other location of R

^{n}, as such a map would not be well-defined. Such a constraint (68) is too strong to handle. Instead, Kantorovich made the following two modifications. First, he transformed the problem into a linear one, by recasting it as follows: Consider the space $\mathcal{P}({\mathbf{R}}^{n}\times {\mathbf{R}}^{n})$of Borel probability measures μ whose push-forwards p

_{1}, p

_{2}on the first and second R

^{n}factors respectively are μ

^{+}and μ

^{−}

_{o}of the Kantorovich functional I

_{K}(75) corresponds to a mapping f

_{o}sought after in Monge’s functional I

_{M}(71).

_{K}∗ as

_{o}is a minimizer of (71). Then Id × f

_{o}: R

^{n}→ R

^{n}× R

^{n}and $\mu ={(\mathrm{Id}\times {f}_{o})}_{\#}{\mu}^{+}\in \mathcal{P}$minimizes I

_{K}in (74) so it is a solution to (75).

^{−}as above. Moreover, let’s assume that M is compact, for simplicity. Let’s choose as cost function

^{+}, μ

^{−}. Naturally, there are many ways to define the discrepancy between measures [71], depending on one’s goals. For our purposes, the optimal transference plan (75) with the cost function (79), in other words the minimizer of (80), enters naturally in the picture. It is interesting to notice that such an optimal transference plan always exists under the above assumptions [35] so the variational problem (75) does have an actual solution. If, in addition one sets

_{1}, μ

_{2}, then it can be immediately checked that W

_{p}(μ

_{1}, μ

_{2}) satisfies all the properties of a distance function. As such, it is called the p- Monge-Kantorovich-Rubinstein-Vasherstein distance, or in short and even though it is a partial misnomer, the p-Wasserstein distance (metric). All these metrics, except W

_{∞}, give the same topology on a compact M which is the weak-∗ topology. For such a distance function to be non-trivial the integral in (82) must converge. This becomes quite important especially when M is non-compact. Hence, the definition of the p-Wasserstein distance restricts the elements of $\mathcal{P}$ to only the ones with finite p-moments. The number of converging moments required for each case of p is the only essential difference between the different W

_{p}metrics on $\mathcal{P}$. The subspace ${\mathcal{P}}_{p}(\mathrm{M}\times \mathrm{M})of\phantom{\rule{0.2em}{0ex}}\mathcal{P}(\mathrm{M}\times \mathrm{M})$of measures with converging p-moments, endowed with the Wasserstein distance then becomes a metric space with several desirable properties [35] and is called the p-Wasserstein space of M. We forego expanding on this topic of current research as further information is not needed in the subsequent discussion.

#### 3.3. The Brenier Map and Its Extensions: the Role of Convexity

^{n}, initially at least, and to assuming that $c(x,y)=\frac{1}{2}{d}^{2}(x,y)$where d denotes the Euclidean distance in R

^{n}. We would like to determine a solution to Monge’s problem in this case, namely to minimize

^{+}and μ

^{−}be probability measures on R

^{n}which are absolutely continuous with respect to the Lebesgue measure (volume) vol of R

^{n}. Then there exists a convex function φ: R

^{n}→ R whose gradient f = ∇φ : R

^{n}→ R

^{n}pushes forward μ

^{+}to μ

^{−}. This map is unique, almost everywhere, and it therefore provides a unique solution to Monge’s problem (70). Immediate generalizations are due to [75]. In the case of the Brenier map, the Monge-Ampére equation (72) for the corresponding densities ρ

^{+}, ρ

^{−}takes the more familiar form

^{n}. The convexity of φ implies that Jacobian obeys det(∇

^{2}φ) ≥ 0 and it also guarantees that f is almost everywhere differentiable. To prove this, one can use the dual Kantorovich formulation (77). The constraint in (76) becomes

_{o}, v

_{o}minimizing (88) and moreover these two functions are Legendre-Fenchel transforms of each other, namely they obey

^{n}to Riemannian manifolds (M, g). This was addressed in part by [77–79] and by [80] for metric measure spaces. One modification that needs to be made is to replace the Legendre-Fenchel transform (86) by the generalized convexity transform

^{n}. Then Brenier’s theorem [78] is extended as follows. For a closed, connected Riemmanian manifold (M, g) let μ

^{+}be absolutely continuous with respect to vol

_{M}and μ

^{−}arbitrary. Then there is a map φ : M → R such that φ

^{cc}= φ (φ is a c-concave function) so that the map exp(−∇φ) pushes forward μ

^{+}to μ

^{−}. Such a map is a unique minimizer within the set $\mathcal{P}(\mathrm{M}\times \mathrm{M})$ for Monge’s transportation functional (83). In the language of the 2-Wasserstein space W

_{2}(M), the Monge transport between μ

^{+}, μ

^{−}takes place along the unique Wasserstein geodesic joining them. The last statement is substantially non-trivial, because if M were a general length space, there could be an uncountably infinite number of geodesics joining the two measures.

^{n}to a Riemannian manifold M. Even then, one can either work directly geometrically with the space at hand, utilising properties of the distance or the volume functions, or express convexity indirectly through functional inequalities such as the Brunn-Minkowski, the Brascamp-Lieb, the Prekopa-Leindler etc. In [79,80] the second approach was taken. Results in Comparison Geometry were derived and extensively used. We will state just one such result to pave the way to the synthetic definition of the generalised Ricci curvature below. To generalise the linear interpolation tx + (1 − t)y between two points x, y ∈ R

^{n}to the case of M with distance function d, they [79] start by considering the set

_{P}(r), and introduce a volume distortion by the volume ratio

_{1}(x, P) = 1, and w

_{t}(x, P) ≥ 1 if the curvature is non-negative. whereas the opposite is true if the curvature is non-positive. In R

^{n}clearly w

_{t}(x, P) = 1. An interpretation of (93) from a relativistic viewpoint, pretending for a moment that we are working in a space of Lorentzian signature, since light does travel on null geodesics w

_{0}(x, P ) represents the magnification of the area of a small light source located near P as is seen by an observer at point x. We compare this distortion function (93) with that of the standard n-dimensional space forms having constant sectional curvatures k > 0 and k < 0 respectively. The Ricci curvature in all these case is Ric = (n − 1)k. Then for k ∈ R

^{n}we set

^{n}, it was noticed that providing lower bounds on the Hessian (63) which is related to the energy functional (57), could be useful in determining rates of convergence of gradient flows on M to their asymptotic configurations. For the “energy” functional (57) with m = 1 [77] calculated its formal Hessian of $\mathcal{P}(\mathrm{M})$and found that it is bounded from below by kg

_{ρ}(56) as long as the generalized Ricci tensor (R

_{N})

_{ij}for N = ∞ of M is bounded from below by kg

_{ij}. Hence the k-convexity, as in (63), of an appropriate energy functional in $\mathcal{P}(\mathrm{M})$ is intimately related to the a lower bound of the generalized Ricci curvature of M. The converse statement, still using the m = 1 case of the “energy” functional in (57), was established by [81]. The obvious question is whether these considerations can be extended to the cases of the “energy” functional in (57) for m ≠ 1 and how would this influence the definitions of the generalized Ricci curvature. An answer was provided by [34,82–86].

#### 3.4. Displacement Convexity and Synthetic Definition of the Generalized Ricci Curvature

_{ν}(μ) by

_{N}: [0, ∞) → R with N ∈ [1, ∞], which is given by

_{ν}with background measure ν is called

- k–displacement convex if for any two ${\mu}_{0},{\mu}_{1}\in \mathcal{P}(\mathrm{X})$and for all Wasserstein geodesics {μ
_{t}}, t ∈ [0, 1], we have$${U}_{v}({\mu}_{t})\le t{U}_{v}({\mu}_{1})+(1-t){U}_{v}({\mu}_{0})-\frac{1}{2}kt(1-t){W}_{2}^{2}({\mu}_{0},{\mu}_{1})$$ - weakly k-displacement convex, if for all ${\mu}_{0},{\mu}_{1}\in \mathcal{P}(\mathrm{X})$there is at least one Wasserstein geodesic along which (108) holds.

_{2}(X) is weakly Λ(U)-displacement convex, namely

_{0}and μ

_{1}

_{0}|x

_{1}), dη(x

_{1}|x

_{0}) indicate the corresponding “conditional” measures. Then, the metric measure space (X, d, ν) has N-Ricci curvature bounded below by k, if there is some optimal transference plan η from μ

_{0}to μ

_{1}with Wasserstein geodesic μ

_{t}, so that for all $U\in {\mathcal{DC}}_{N}$ and for all t ∈ [0, 1]

_{2}(M) are used to reflect the Ricci curvature properties of M, all in a comparison sense. We observe that the definition (115) is synthetic: nowhere have we required the metric measure space to be smooth. Its apparent drawback for applications to Physics is that this definition of Ricci curvature is only defined in a comparison sense. However, it is directly related to the convexity properties of ${\mathcal{S}}_{q}$ in the Wasserstein space W

_{2}(X). On the other hand, the definition (41) of the generalised Ricci tensor/curvature in smooth metric measure spaces is local, so it is easy to compute, in principle, but it appears to have nothing to do with ${\mathcal{S}}_{q}$. However (41) and (115) give the same result, in a comparison sense, for a measured length space, hence for a Riemannian manifold: The Riemannian manifold (M, g) has its generalised Ricci curvature bounded below by k (115) if and only if [86] its N-Ricci tensor (41) is also bounded below by k, namely (R

_{N})

_{ij}≥ kg

_{ij}. Before closing this Section, it may be worth noticing that for non-branching spaces such as Riemannian manifolds, there is no real distinction between an element of the displacement convexity class ${\mathcal{DC}}_{N}$ and (97), so in this sense the entropic functional ${\mathcal{S}}_{q}$ itself is unique in the determination of the generalised Ricci curvature (115).

## 4. Isoperimetric Interpretation of the Non-extensive Parameter and Related Matters

^{23}which represents a gross mismatch with the values of N computed through (116) and from data fittings of q. We have not been able to resolve this interpretational conundrum in an acceptably convincing way. The best that we can currently state, is to vaguely suggest that N should actually represent an effective isoperimetric dimension of the configuration or phase space per degree of freedom, something more akin to $\frac{N}{n}$ in our notation. Given this, it might be of some interest to attempt to explain the appearance of the escort distributions [6,95] used in the calculations of the thermodynamic parameters [6], under this light.

_{N}> 0 then, according to (43), the corresponding measures on the configuration / phase space increase slower than a power-law fashion with exponent equal to N. This also provides another dimensional interpretation of N as the maximal Hausdorff dimension exponent bounding the expansion of measures in the configuration or phase space of the microscopic system.

_{#}of dvol

_{M}is just a multiple of dvol

_{B}and let N = dim F. Let the Ricci curvatures of M, B be indicated by superscripts with respect to their corresponding metrics. Then for any k ∈ R, [49] proves that if Ric

^{M}≥ k, then $Ri{c}_{N}^{\mathrm{B}}\ge k$. So, a way to understand the meaning of the generalized Ricci tensor is to see it as the Ricci tensor due to the submersion of a higher dimensional space preserving the measure of the base up to a multiplicative constant. This statement may also make the generalised Ricci tensor quite useful for applications in theories involving higher (greater than 4) dimensional space-times. The result and the general ideas are also close to the treatment of [99,100] who expressed the non-extensive parameter q in terms of the scaling properties of the Hamiltonians of the “thermostat” F and of the system under study B.

## 5. Assessment and Omissions

## Acknowledgments

## Conflicts of Interest

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Kalogeropoulos, N.
Ricci Curvature, Isoperimetry and a Non-additive Entropy. *Entropy* **2015**, *17*, 1278-1308.
https://doi.org/10.3390/e17031278

**AMA Style**

Kalogeropoulos N.
Ricci Curvature, Isoperimetry and a Non-additive Entropy. *Entropy*. 2015; 17(3):1278-1308.
https://doi.org/10.3390/e17031278

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Kalogeropoulos, Nikos.
2015. "Ricci Curvature, Isoperimetry and a Non-additive Entropy" *Entropy* 17, no. 3: 1278-1308.
https://doi.org/10.3390/e17031278