The problem of camouflaging via mirror reflections

1. Introduction

The idea of invisibility has always been attractive for people. Stories on magic cap and cloak of invisibility form an essential part of folklore, myths and fairy tales. Methods of camouflaging establishments, troops and other objects of importance are of great interest to the military at all times; one of the most famous developments of the twentieth century in this area is the stealth technology aiming at making airplanes invisible to enemy radar.

In the last decades, intensive work has been carried out on developing technology of meta-materials possessing unusual properties (e.g. [1]), having in mind, in particular, creating something like a transparent meta-material cover with varying refractive index that makes invisible every object placed inside.

An important and interesting mathematical construction in the two-dimensional case is proposed in the paper by Leonhardt [2]. It describes how to make an object invisible by wrapping a lens (a transparent material with varying refractive index) around it.

There is an interesting question: to what extent can one create the effect of invisibility, if only mirror systems are allowed to be used? Mirrors are much easier and cheaper for fabrication than hypothetical meta-material structures, and even than traditional lenses with controlled refractive index. Some results in this direction have already been obtained. There exist and are described (connected) bodies invisible from one point [3,4] and (infinitely connected) bodies invisible from two points [5]. There exist (connected and even simply connected) bodies invisible in one direction (that is, from an infinitely distant point) [6], (finitely connected) bodies invisible in two directions [7], as well as (infinitely connected) bodies invisible in three [8] and (in the two-dimensional case) in n-directions, where the number $n\in \mathbb{N}$ of directions is arbitrary [9].

On the other hand, there are negative results revealing restricted possibilities of mirror systems as compared with more sophisticated technologies. In particular, non-existence of perfectly invisible bodies (that is those that are invisible in any direction or (equivalently) from any point outside the body) is proved in [7]. Further, a conjecture proposed in [3] states that the set of light rays that are invisible for any fixed body has measure zero. This conjecture is closely connected with the long-standing Ivrii’s conjecture [10] stating that the measure of the set of periodic billiard trajectories in a bounded domain has measure zero. If Ivrii’s conjecture is true then, most probably, true also is the conjecture on invisible light rays.

At the moment Ivrii’s conjecture has been proved only for trajectories with three [11,12] and four [13,14] reflections. The corresponding invisibility conjecture for the case of three reflections is easily obtained by slightly rephrasing the proof of Ivrii’s conjecture with three reflections. A unified approach developed by Glutsyuk in [13] based on complexification of billiards allows one to derive both Ivrii’s and invisibility conjectures in the case of four reflections from his theorem of classification of four-reflective complex planar analytic billiards (Theorem 1.7 in [13]). Note that the proof of the conjectures in the case of four reflections is much more difficult than the case of three reflections.

In real life, quite common is the situation when perfect invisibility is impossible to achieve. In such cases, one tries to reach the effect of partial invisibility, or camouflaging, when the object, though not disappearing completely, still becomes difficult to detect by an observer. It is natural to set such a question in the framework of mirror invisibility. In order to state a mathematical problem, one needs first to determine an index of visibility, a certain positive quantity which is close to zero if the body is, in a sense, difficult to detect. This quantity should never vanish, since perfectly invisible bodies do not exist.

Then one should consider the question, how small can this index be made in a certain class of bodies? For example, if even the index does not vanish, is it possible to construct a sequence of bodies of constant volume with the index going to zero?

Choosing the visibility index is not an easy task; it is more difficult than just defining the notion of invisibility. The body is observed against a certain background, and the choice will depend, in particular, on the distance of the body from the background. In the limit, when the background is infinitely distant, the visibility index is determined by the angles of deviation of light rays from their original directions and does not depend on transverse displacement of the rays. This limit will be used later on in this paper.

The aim of the paper is to give (partial) answers to the questions stated above. If the body has volume A and is contained in a sphere of radius r, then its visibility index is not less than a certain positive value, a function of A and r. This function goes to zero when A is constant as $r\to \mathrm{\infty }$.

2. Main definitions and statement of the results

First of all fix the notation. A body with specular surface is a bounded finitely connected domain with piecewise smooth boundary in Euclidean space ${\mathbb{R}}^d$ with d≥2. It will be called a domain and designated by D. Since everything is about specular reflections in the framework of geometric optics, we adopt the notation of billiard theory and consider the billiard in ${\mathbb{R}}^d\setminus D$.

Fix a domain D and take a sphere $S_R^{d-1}$ of radius R>0 centred at the origin and containing D. It is assumed that the background lies on the sphere. As a result of observation of the background one must conclude whether the body is or is not present here. For a point ξ on the sphere and a unit vector v such that 〈v,ξ〉<0 consider the trajectory of a billiard particle starting at ξ with the velocity v and the half-line with the endpoint ξ and the directing vector v, and denote by θ=θ_R,D(v,ξ) the angular distance between the (second) points of intersection of the trajectory and of the half-line with the sphere (figure 1).

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We define the measure spaces

equipped with the measure μ_R=μ defined by dμ(v,ξ)=|〈v, n(ξ)〉| dv dξ, where n(ξ)=ξ/R is the outer unit normal to the sphere $S_R^{d-1}$ at the point ξ. Note that the spaces ${(S^{d-1}\times S_R^{d-1})}_{-}$ and ${(S^{d-1}\times S_R^{d-1})}_{+}$ correspond to billiard trajectories entering the sphere $S_R^{d-1}$ and leaving it, respectively, and μ is a natural measure counting the amount of billiard trajectories intersecting the sphere.

Take a monotone increasing function $f:[0,\hspace{0.17em}\pi ]\to \mathbb{R}$ such that f(0)=0 and consider the value

which will be called the visibility index of D.

Note that the equality ${\mathcal{F}}_R(D)=0$ does not yet guarantee invisibility of D. In fact, the domain D is invisible, if and only if ${\mathcal{F}}_R(D)=0$ for any R sufficiently large.

In the limit $R\to \mathrm{\infty }$ the quantity θ does not depend on the transverse displacement (shift) of the trajectory going away, but only on the angle between the initial and final velocities. Let us introduce some more notation. For the billiard trajectory entering the sphere $S_R^{d-1}$ at a point ξ and having a velocity v at this point, we denote by

the second point of intersection of this trajectory with the sphere (when leaving the sphere $S_R^{d-1}$) and its velocity at this point. (Note that the velocity v⁺ does not depend on the radius R of the sphere containing D.) The mapping is a one-to-one mapping from ${(S^{d-1}\times S_R^{d-1})}_{-}$ onto ${(S^{d-1}\times S_R^{d-1})}_{+}$ (defined up to a subset of measure zero) preserving the measure μ.

Then in the limit mentioned above, θ is the angle between v and v⁺, $\theta =\arccos ⟨v,v^{+}⟩$, and one comes to the following formula for the visibility index, $\mathcal{F}(D)=\underset{R\to \mathrm{\infty }}{lim}{\mathcal{F}}_R(D)$:

with r being taken sufficiently large. The value of the integral in this formula does not depend on r (for the proof see Proposition 1.1 of ch. 1 in [3]). The visibility index in this case is related to the situation when the distance to the background is much greater than the size of the domain itself.

Denote by s_d−1=|S^d−1|=2π^d/2/Γ(d/2) the area of the (d−1)-dimensional unit sphere, and by b_d=2π^d/2/dΓ(d/2) the volume of the d-dimensional ball. One has, in particular, s₀=2, s₁=2π, s₂=4π, b₁=2, b₂=π, b₃=4π/3.

Assume that

and introduce the notation where B stands for the beta function, B(x,y)=Γ(x)Γ(y)/Γ(x+y). In particular, in the three-dimensional case we have c₃=c(k^k/(k+1)^k+1)(1/(2π)^k−1).

Theorem 2.1 establishes a connection between the visibility index of a domain, its volume and the radius of a ball containing this domain.

Theorem 2.1

Let a domain$D\subset {\mathbb{R}}^d$be contained in a ball of radius r. Then its visibility index$\mathcal{F}(D)$, its volume |D| and r are related by the inequality

where h_dis a function of a positive variable satisfying

Note that the values $\mathcal{F}(D)/r^{d-1}$ and |D|/r^d are preserved under a scaling transformation (applied to both D and the ambient ball). It is natural therefore that theorem 2.1 relates these two values.

Remark 2.2

It follows from theorem 2.1 that

This means that the infimum of visibility index in a class of domains with fixed volume contained in a certain ball is greater than a positive constant. This constant goes to zero when the radius of the ambient ball tends to infinity. It would be interesting to learn something about the upper bound for this infimum, at least for a special kind of the function f defining the visibility index. In particular, does the infimum go to zero as $r\to \mathrm{\infty }$ (or, equivalently, as the diameter of D goes to infinity)? In other words, is it possible to construct a sequence of domains with fixed volume and with the visibility index going to zero?¹ We do not know the answer to this question.

One may wish to have a more direct estimate of the visibility index (without the term o(1)). We shall derive such estimates in the two- and three-dimensional cases for a particular choice of the function f. Namely, take $f(\theta )=1-\cos \hspace{0.17em}\theta$; the resulting visibility index

(with r taken sufficiently large) has a simple mechanical interpretation in the framework of Newtonian aerodynamics [15]: it is just the mean value over all v of the aerodynamic resistance of D in the direction of v.² We shall call it the mean resistance of D.

Note that the mean resistance of a convex domain $C\subset {\mathbb{R}}^d$ can easily be determined; denoting by |∂C| the (d−1)-dimensional area of its boundary, one has $\mathfrak{F}(C)=(4/(d+1))b_{d-1}|\mathrm{\partial }C|$. In the two- and three-dimensional cases one has, respectively, $\mathfrak{F}(C)=8|\mathrm{\partial }C|/3$ and $\mathfrak{F}(C)=\pi |\mathrm{\partial }C|$. In particular, the mean resistances of the two- and three-dimensional balls, $B_r^2$ and $B_r^3$, of radius r are equal, respectively, to $\mathfrak{F}(B_r^2)=\frac{16}{3}\pi r$ and $\mathfrak{F}(B_r^3)=4{\pi }^2{r}^2$ (see §§6.1.1 and 6.2 of [3] for details).

The following formulae for the mean resistance in the two- and three-dimensional cases are obtained from theorem 2.1 by direct substitution c=1/2, k=2,

The following theorem allows one to get rid of the term o(1) in the above formulae.

Theorem 2.3

(a) Let a planar domain D with the area |D| be contained in a circle of radius r. Then

(b) Let a three-dimensional domain D with the volume |D| be contained in a ball of radius r. Then

It is instructive to rewrite these formulae in terms of reduced volume κ_D and reduced resistance ${\hat{\mathfrak{F}}}_D$ defined by

where r is the radius of the smallest ball containing D. One always has 0<κ_D≤1, and κ_D=1 iff D is a ball. In the latter case, we have ${\hat{\mathfrak{F}}}_D=1$. In the case d=2, one has κ_D=|D|/(πr²) and ${\hat{\mathfrak{F}}}_D=\mathfrak{F}(D)/(\frac{16}{3}\pi r)$, and in the case d=3, one has ${\kappa }_D=|D|/(\frac{4}{3}\pi r^3)$ and ${\hat{\mathfrak{F}}}_D=\mathfrak{F}(D)/(4{\pi }^2{r}^2)$.

It is interesting to note that

for all 0<κ<1. This can easily be derived from Theorem 6.2 in ch. 6 of [3] by taking a sequence of domains inscribed in a certain ball and having the property of asymptotically perfect retro-reflection (i.e. the initial velocity of the most part of incident particles is reversed as a result of reflections). Note that one can always ensure that the volume of each domain in the sequence equals κ by taking off the domain an appropriate smaller concentric ball.

On the contrary, only rough estimates are known for the infimum of ${\hat{\mathfrak{F}}}_D$. In particular, the following estimates in the two- and three-dimensional cases follow directly from theorem 2.3,

These estimates are far from being sharp. Indeed, from the same Theorem 6.2 in [3] one can derive the exact value of the lower limit of ${\hat{\mathfrak{F}}}_D$ when κ_D→1; in particular,

On the other hand, the values of the lower limits given by formulae (2.4) are much smaller, $\underset{\tau \to 1^{-}}{lim}\underset{{\kappa }_D=\tau }{inf}{\hat{\mathfrak{F}}}_D\ge {\pi }^3/288\approx 0.11$ for d=2 and $\underset{\tau \to 1^{-}}{lim}\underset{{\kappa }_D=\tau }{inf}{\hat{\mathfrak{F}}}_D\ge 16/729\approx 0.022$ for d=3.

There is a question on a natural generalization of formula (2.4). Consider the relative volume of a domain in its convex envelope, and let the normalized resistance be chosen so that the resistance of the convex envelope of the body equals 1. Is it possible to derive a sensible estimate for the normalized resistance in the spirit of formula (2.4)?

Remark 2.4

The statement of theorem 2.3 can also be interpreted in terms of Newton’s problem of minimal resistance [15]. Consider a body moving in a rarefied medium of point particles. The medium is so rare that mutual interaction of particles is neglected, and particles are reflected elastically when hitting the body’s boundary. One needs to find a body, from a prescribed class of bodies, that has the smallest aerodynamic resistance. There has been a significant progress in this problem in 1990s and 2000s (e.g. [3,16–19]).

Suppose now that the body D translates in the medium and at the same time rotates (somersaults) very slowly and chaotically. In this case, one is interested in minimizing the mean value of its resistance in all possible directions, i.e. the value $\mathfrak{F}(D)$. Theorem 2.3 gives a lower estimate of this value in a class of bodies with fixed volume.

Remark 2.5

The statements of theorems 2.1 and 2.3 hold for broader classes of dynamical systems than billiards. It suffices that the system satisfies the following conditions:

• (i) the motion is free outside a sphere of radius r;

• (ii) all the trajectories of the system are continuous curves;

• (iii) the standard measure dv dx is invariant under the dynamics of the system.

In this case, the proofs (given in the next section) go through without change.

One can, for example, take the free dynamics outside $D\subset {\mathbb{R}}^2$ with a pseudo-billiard law of reflection off the boundary ∂D; this law is induced by any one-to-one mapping of the segment [−π/2, π/2] onto itself preserving the measure $\cos \hspace{0.17em}\phi \hspace{0.17em}\mathrm{d}\phi$.

Remark 2.6

Taking account of Jung’s inequality between the diameter diam(D) of a set D and the smallest radius r=r(D) of a ball containing D,

(e.g. subsection 2.6 in [20]), the inequalities in theorem 2.3 can be replaced by the following (slightly weaker) ones, in the two-dimensional case and in the three-dimensional case. Substituting the exact values with approximate ones, one can then write down $\mathfrak{F}(D)>0.9\hspace{0.17em}|D{|}^3\text{diam}{(D)}^{-5}$ in the two-dimensional case and $\mathfrak{F}(D)>0.36\hspace{0.17em}|D{|}^3\text{diam}{(D)}^{-7}$ in the three-dimensional case.

Further, using Jung’s inequality and the obvious relation diam(D)≤2r, the inequality in theorem 2.1 can be replaced with

where h_d is as in theorem 2.1.

3. Proofs of the theorems

All statements below are true up to subsets of measure zero.

Let us first prove theorem 2.1. We consider the billiard inside the ball $B_r^d$ of radius r and outside D. A particle starts moving at a point $\xi \in S_r^{d-1}$ and with a velocity v∈S^d−1 directed inside the sphere S^d−1, then makes several reflections off D, and finally intersects S^d−1 again (at the point ξ⁺ and with the velocity v⁺) and disappears at the moment of intersection. The phase space of the billiard is $S^{d-1}\times (B_r^d\setminus D)$, and its volume V (with respect to the standard Liouville measure dv dx) equals

Denote by l_D,r(v,ξ) the length of the billiard trajectory with the initial data $(v,\xi )\in {(S^{d-1}\times S_r^{d-1})}_{-}$ until the final intersection with $S_r^{d-1}$. We use Santaló–Stoyanov formula (e.g. [21,22]), which in our case states that the phase volume is greater than or equal to the integral of the length of billiard trajectories over the initial data,

(note that the sign ‘≥’ in this formula is due to the fact that a part of the phase space may be inaccessible for particles starting at the ambient sphere $S_r^{d-1}$). Writing ξ⁺ in place of ξ⁺_D,r(v,ξ) for brevity and using the obvious inequality l_D,r(v,ξ)≥|ξ−ξ⁺|, we then obtain

Further, take an orthonormal coordinate system, x=(x₁,…x_d), centred at the origin and denote by ${\mathcal{R}}_v$ the rotation of ${\mathbb{R}}^d$ about the origin such that

• (a) under this rotation, v goes to $(\overline{0},\hspace{0.17em}1):=(0,\dots ,0,1)$; that is, ${\mathcal{R}}_{v}v=(\overline{0},1)$;

• (b) the two-dimensional subspace of ${\mathbb{R}}^d$ spanned by the vectors v and $(\overline{0},1)$ is invariant under ${\mathcal{R}}_v$;

• (c) ${\mathcal{R}}_v$ acts as identity on the orthogonal complement to this subspace.

Denote the upper and lower hemispheres of radius r by

Denote η′=(η₁,…,η_d−1) and consider the standard measure dη′ on both the hemispheres (that is, the measure of a Borel subset in ${\mathcal{S}}_r^{\pm }$ is equal to the Lebesgue measure of its orthogonal projection on the subspace η_d=0). For each choice of the sign ‘+’ or ‘−’ and for all v∈S^d−1, the mapping $\xi \mapsto {\mathcal{R}}_v\xi$ from the hemisphere $\{\xi \in S_r^{d-1}:\pm ⟨v,\hspace{0.17em}\xi ⟩\ge 0\}$ with the measure |〈v,n(ξ)〉| dξ onto the hemisphere ${\mathcal{S}}_r^{\pm }$ with the measure dη′ is measure preserving.

It follows (by Cavalieri’s principle) that the bijective mapping $(v,\xi )\mapsto (v,{\mathcal{R}}_v\xi )$ between the space ${(S^{d-1}\times S_r^{d-1})}_{\pm }$ with the measure μ (recall that it is defined by dμ(v,ξ)=|〈v, n(ξ)〉| dv dξ) and the space $S^{d-1}\times {\mathcal{S}}_r^{\pm }$ with the measure defined by dv dη′ is also measure preserving. This implies, in particular, that

Now define the measures μ_± on ${\mathcal{S}}_r^{\pm }$ by dμ_±=s_d−1 dη′; then the mappings ${\pi }_{\pm }:{(S^{d-1}\times S_r^{d-1})}_{\pm }\to {\mathcal{S}}_r^{\pm }$ defined by

are measure preserving.

We have

Fix a value 0<ϕ<π and let Σ_ϕ be the set of values $(v,\xi )\in {(S^{d-1}\times S_r^{d-1})}_{-}$ such that the angle between v and v⁺=v⁺_D(v,ξ) is greater or equal than ϕ, that is,

From (3.2) and (3.3), taking into account that $|{\mathcal{R}}_v{\xi }^{+}-{\mathcal{R}}_{v^{+}}{\xi }^{+}|\le 2r$, we get

Let us estimate the three terms in the right-hand side of (3.5).

(a) Denote for brevity $X={(S^{d-1}\times S_r^{d-1})}_{-}$ and recall that f(ϕ) is positive and monotone increasing for 0<ϕ<π. By Chebyshev’s inequality for t>0 one has

Substituting t=f(ϕ), using the definitions (2.1) and (3.4), and multiplying both parts of the inequality by 2r one obtains

If $(v,\xi )\in {(S^{d-1}\times S_r^{d-1})}_{-}\setminus {\mathit{\Sigma }}_{\varphi }$ then the angle between v and v⁺ is less than ϕ, and denoting by α=α(v) and α⁺=α(v⁺) the angles formed by the vectors v and v⁺ with $(\overline{0},1)$, 0≤α, α⁺≤π; we have |α−α⁺|≤ϕ.

(b) In the case d=2, one obviously has

In the case d≥3, the estimate is more difficult.

Consider the three-dimensional subspace of ${\mathbb{R}}^d$ spanned by the vectors v, v⁺ and $(\overline{0},1)$. The restrictions of ${\mathcal{R}}_v$ and ${\mathcal{R}}_{v^{+}}$ on this subspace are rotations by the angles α and α⁺, respectively. Let w and w⁺ be unit vectors in this subspace pointing at directions of the rotation axes. Both w and w⁺ are orthogonal to $(\overline{0},1)$. The restriction of ${\mathcal{R}}_{v^{+}}^{-1}{\mathcal{R}}_v$ on this subspace acts as a rotation by an angle β, and its restriction on the orthogonal complement to this subspace is an identity. We have

therefore, we need to estimate $\sin \hspace{0.17em}(\beta /2)$. To that end we shall proceed to some trigonometric calculations.

Introduce an orthonormal coordinate system x, y, z in the chosen subspace, where the third coordinate axis coincides with the dth axis of the original space ${\mathbb{R}}^d$ and the origin coincides with the origin in the space ${\mathbb{R}}^d$. In this system, the coordinate vectors v, v⁺ and $(\overline{0},1)$ take the form

One has

Further, one easily finds that $w=(-\sin \hspace{0.17em}\theta ,\hspace{0.17em}\cos \hspace{0.17em}\theta ,\hspace{0.17em}0)$, $w^{+}=(-\sin \hspace{0.17em}{\theta }^{+},\hspace{0.17em}\cos \hspace{0.17em}{\theta }^{+},\hspace{0.17em}0)$, and therefore Taking into account that $⟨v,v^{+}⟩\ge \cos \hspace{0.17em}\varphi$ and using (3.7) and (3.8), one finds

In what follows, we shall use the same notation ${\mathcal{R}}_v$ and ${\mathcal{R}}_{v^{+}}$ for the restrictions of the corresponding rotations on our three-dimensional subspace. It is convenient to represent them in the quaternionic form: ${\mathcal{R}}_v$ is the action u↦quq⁻¹ of the quaternion

and ${\mathcal{R}}_{v^{+}}$ is the action u↦q₊uq⁻¹₊ of the quaternion Correspondingly, ${\mathcal{R}}_{v^{+}}^{-1}{\mathcal{R}}_v$ is the action of the quaternion which has the real part From this formula, using (3.9) and taking into account the double angle formulae for sine and cosine, one obtains the estimate Denoting $\cos \hspace{0.17em}({\alpha }^{+}/2)=:z$, one comes to the inequality If 0<α<π−ϕ, the infimum of the expression in the brackets is attained at $z=\sqrt{\cos \hspace{0.17em}\varphi +\cos \hspace{0.17em}\alpha }/\sqrt{2}$. Substituting this value in the latter inequality, one gets From here after some algebra one finally obtains Note that the right-hand side in this inequality is smaller than 1.

Thus, we have

where in the case d≥3, and $h_{\varphi }(\alpha )=\sin \hspace{0.17em}(\varphi /2)$ in the case d=2. We now have an estimate for the second integral in the right-hand side of (3.5)

We now need to estimate the integral in the right-hand side of this inequality. Integrating by ξ gives us the factor b_d−1r^d−1. Integrating by v over S^d−1 amounts to integration with the differential $s_{d-2}{\sin }^{d-2}\alpha \hspace{0.17em}\mathrm{d}\alpha$ over the interval α∈[0,π]. Thus, we get

in the case d=2 and in the case d≥3.

Introducing the functions ${\mathcal{I}}_d(\varphi ),\hspace{0.17em}\varphi \in [0,\hspace{0.17em}\pi ],\hspace{0.17em}d=2,3,\dots$ by

and for d≥3, we can now write For small values of ϕ, we have the following asymptotic behaviour: The function ${\mathcal{I}}_3$ can easily be calculated in the case d=3,

(c) Now consider the first term in the right-hand side of (3.5).

Recall that the mapping $T=T_{D,r}:{(S^{d-1}\times S_r^{d-1})}_{-}\to {(S^{d-1}\times S_r^{d-1})}_{+}$ is defined by T(v,ξ)=(v⁺_D(v,ξ),ξ⁺_D,R(v,ξ)). It preserves the measure μ, and therefore induces a measure on ${(S^{d-1}\times S_r^{d-1})}_{-}\times {(S^{d-1}\times S_r^{d-1})}_{+}$ concentrated on the graph of T and whose projections on ${(S^{d-1}\times S_r^{d-1})}_{-}$ and ${(S^{d-1}\times S_r^{d-1})}_{+}$ coincide with μ. The push forward of this measure under the map³${\pi }_{-}\times {\pi }_{+}:{(S^{d-1}\times S_r^{d-1})}_{-}\times {(S^{d-1}\times S_r^{d-1})}_{+}\to {\mathcal{S}}_r^{-}\times {\mathcal{S}}_r^{+}$ (let it be denoted by ν_D,r) is a measure on ${\mathcal{S}}_r^{-}\times {\mathcal{S}}_r^{+}$ whose projections on ${\mathcal{S}}_r^{-}$ and on ${\mathcal{S}}_r^{+}$ coincide, respectively, with μ₋ and μ₊.

Therefore, we have

where the infimum is taken over all measures ν whose projections on ${\mathcal{S}}_r^{-}$ and on ${\mathcal{S}}_r^{+}$ coincide with μ₋ and μ₊. This problem of searching for the minimum is actually a problem of optimal mass transport, which in this case is easy to solve. We use the inequality $|\eta -{\eta }^{+}|\ge |{\eta }_d-{\eta }_d^{+}|={\eta }_d^{+}-{\eta }_d$ (since ${\eta }_d=-\sqrt{r^2-\sum _{i=1}^{d-1}{\eta }_i^2}\le 0$ and ${\eta }_d^{+}=\sqrt{r^2-\sum _{i=1}^{d-1}{({\eta }_i^{+})}^2}\ge 0$) to get

⁴That is, we have

(d) From (3.1), (3.5), (3.6), (3.11) and (3.13), we obtain

It follows that

Using asymptotic formulae (2.2) and (3.12) for f and ${\mathcal{I}}_d$, respectively, and replacing both terms in the right-hand side of (3.14) with their approximated values (as ϕ→0⁺), we obtain the expression

where The minimum of (3.15) is equal to ((k+1)/k)α^1/(k+1)β^k/(k+1) and is attained at ϕ_*=(α/β)^1/(k+1). Substituting this value ϕ_* in the right-hand side of (3.14) and raising both parts of the resulting inequality to the (k+1)th power, one obtains where c_d is defined by (2.3) and o(1) means a function of $\mathcal{F}(D)/r^{d-1}$ vanishing when its argument goes to zero. Reversing relation (3.16), one gets this time o(1) means a function of |D|/r^d vanishing when its argument goes to zero. Theorem 2.1 is proved.

Let us now prove theorem 2.3. Here we have $f(\varphi )=1-\cos \hspace{0.17em}\varphi$ and use the notation $\mathfrak{F}$ in place of $\mathcal{F}$ in this particular case.

If d=2, substitute s₁=2π, b₁=2, s₀=2 and ${\mathcal{I}}_2(\varphi )=\pi \sin \hspace{0.17em}(\varphi /2)$ into (3.14) to obtain

Introducing the shorthand notation $z=\sin \hspace{0.17em}(\varphi /2),\hspace{0.17em}A=|D|,\hspace{0.17em}\mathfrak{F}=\mathfrak{F}(D)$, this inequality can be rewritten as

Our goal is to prove the inequality

which is equivalent to statement (a) of theorem 2.3.

Consider two cases. If $\mathfrak{F}\le 4\pi \hspace{0.17em}r$, the infimum in (3.17) is attained at $z_{\ast }={(\mathfrak{F}/(4\pi \hspace{0.17em}r))}^{1/3}$, and substituting z_* in (3.17), we get (3.18). On the other hand, if $\mathfrak{F}>4\pi \hspace{0.17em}r$, we obviously have (since D is contained in a circle of radius r)

and we again come to (3.18). Thus, statement (a) of theorem 2.3 is proved.

If d=3, one has s₂=4π, b₂=π, s₁=2π and ${\mathcal{I}}_3(\varphi )=4\sin \hspace{0.17em}(\varphi /2)-2{\sin }^2(\varphi /2)$, and inequality (3.14) takes the form

Introducing the notation $\tilde{A}=|D|/(2\pi \hspace{0.17em}{r}^3),\hspace{0.17em}\tilde{\mathfrak{F}}=\mathfrak{F}(D)/(8{\pi }^2{r}^2)$ and $z=\sin \hspace{0.17em}(\varphi /2)$, one rewrites the last inequality in the form

We are going to prove the inequality

which is equivalent to statement (b) of theorem 2.3.

After a simple algebra one concludes that if $\tilde{\mathfrak{F}}>27/256$, we have h′(z)<0 for all z>0, has a unique zero z=3/4. If $0<\tilde{\mathfrak{F}}\le 27/256$, the equation h′(z)=0 has two positive zeros (coinciding when $\tilde{\mathfrak{F}}=27/256$). The smallest zero $z_{\ast }=z_{\ast }(\tilde{\mathfrak{F}})$ (which is a local minimizer of h if $\mathfrak{F}$ is strictly smaller than 27/256) satisfies the inequality 0<z_*≤3/4. It is also straightforward to check that

Consider two cases. If $\tilde{\mathfrak{F}}\le 27/256$, we substitute z=z_* in (3.19) and use (3.21) to obtain $\tilde{A}\le 3z_{\ast }(1-2z_{\ast }/3)$. Taking the third power of both sides of this inequality and using that (1−2z/3)³<1−z for 0<z≤3/4, we come to (3.20). If, otherwise, $\tilde{\mathfrak{F}}>27/256$, we use that $|D|\le \frac{4}{3}\pi r^3$, and therefore $\tilde{A}\le 2/3$. It follows that ${\tilde{A}}^3\le 8/27<27\cdot 27/256<27\tilde{\mathfrak{F}}$, and (3.20) again follows. Thus, statement (b) of theorem 2.3 is also proved.

Data accessibility

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Competing interests

I declare I have no competing interests.

Funding

This work was supported by Portuguese funds through CIDMA—Center for Research and Development in Mathematics and Applications and FCT—Portuguese Foundation for Science and Technology, within the project UID/MAT/04106/2013.

Footnotes