We study the expectation of the number of vertices f0(K(n))and the missed areaA(K\Kn)asntends to infinity

ON RANDOM APPROXIMATIONS BY GENERALIZED DISC-POLYGONS

FERENC FODOR, DÁNIEL I. PAPVÁRIANDVIKTOR VÍGH

Abstract. For two convex discsK andL, we say thatK isL-convex (Lángiet al.,Aequationes Math.85(1–2) (2013), 41–67) if it is equal to the intersection of all translates ofLthat containK. In L-convexity, the setLplays a similar role as closed half-spaces do in the classical notion of convexity.

We study the following probability model: LetKandLbeC₊² smooth convex discs such thatKis L-convex. Selectnindependent and identically distributed uniform random pointsx1, . . . ,xnfromK, and consider the intersectionK₍n)of all translates ofLthat contain all ofx1, . . . ,xn. The setK₍n)is a randomL-convex polygon inK. We study the expectation of the number of vertices f0(K₍n))and the missed areaA(K\Kn)asntends to infinity. We consider two special cases of the model. In the first case, we assume that the maximum of the curvature of the boundary ofLis strictly less than 1 and the minimum of the curvature ofKis larger than 1. In this setting, the expected number of vertices and missed area behave in a similar way as in the classical convex case and in ther-spindle convex case (whenLis a radiusrcircular disc), see (Fodoret al.,Adv. in Appl. Probab.46(4) (2014), 899–

918). The other case we study is whenK=L. This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only onL. This was previously observed in the special case whenLis a circle of radiusrin Fodoret al.(Adv. in Appl.

Probab.46(4) (2014), 899–918). We also determine the extrema of the limit of the expectation of the number of vertices ofL₍n)ifLis a convex discs of constant width 1. The formulas we prove can be considered as generalizations of the correspondingr-spindle convex statements proved by Fodor et al.in (Adv. in Appl. Probab.46(4) (2014), 899–918).

§1. Introduction and results. Rényi and Sulanke started the investigation of the asymptotic properties of random polytopes in their seminal papers [22–24]. They studied the planar version of the following probability model: LetKbe a convex body (compact convex set with interior points) in Euclideand-spaceR^d, and selectnindependent and identically distributed random pointsx1, . . . ,xnfromKaccording to the uniform probability distribution. The convex hull of the random pointsx₁, . . . ,x_n is a (random) polytopeK_ninK, which tends toK with probability 1 asn→ ∞. Common random variables associated with such polytopes are, for example, the number ofi-dimensional faces fori=0, . . . ,d−1, and the difference of thejth intrinsic volumes ofK andK_n for j =1, . . . ,d−1. After the works of Rényi and Sulanke, many of the results in the theory of random polytopes have been of asymptotic type, meaning that they describe the limiting behaviour of some aspect, such as expectation and variance, of a random variable as the number of pointsntends to infinity. Our motivations come, in part, from the asymptotic formulas proved by Rényi and Sulanke for the expected number of vertices [22, Satz 3, p. 83] and the missed area [23, Satz 1 (48), p. 144] of random convex polygons in sufficiently smooth convex discs. Our aim is to prove similar statements in a different, and somewhat more general, setting in the Euclidean plane. In the last few decades,

Received 3 July 2019, published online 9 April 2020.

MSC (2010): 52A22 (primary), 52A27, 60D05 (secondary).

The researchers were supported by the Ministry of Human Capacities, Hungary grant 20391-3/2018/FEKUSTRAT. The first and the third authors were also supported by Hungarian National Research, Development and Innovation Office NKFIH grant K 116451.

We note that parts of this paper appeared in the Bachelor’s thesis [20] of Papvári.

© 2020 The Authors.Mathematikapublished by John Wiley & Sons Ltd on behalf of University College London. This is an open access article under the terms of theCreative Commons AttributionLicense, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

the literature on this topic has grown enormously, especially in the generald-dimensional setting. We do not venture to give an overview of the subject in this paper, instead, we refer to the comprehensive surveys [1, 15, 21, 26–30] for more information and references.

Recently, another probability model of random polytopes emerged that is based on intersections of congruent closed balls of suitable radius, see [9, 10, 13]. For a fixedr >0, a convex discK ⊂R²is calledr-spindle convex (sometimes also calledr-hyperconvex [8]

orr-convex [6, 7]) if, together with any two pointsx,y∈K, the set [x,y]_r, consisting of all shorter circular arcs of radius at leastr and connectingxandy, is contained inK. One can also think of [x,y]_r as the intersection of all radiusrclosed circular discs that containxand y. In this concept, the set [x,y]_r plays a similar role as the segment in the classical notion of convexity. The intersection of a finite number of radiusr circles is called a disc polygon of radiusr. The concept of spindle convexity emerged from a paper of Mayer [18], and has subsequently been investigated from different points of view. For more information on spindle convex sets and further references, we refer to the paper by Bezdeket al.[2] and the recent book by Martiniet al.[17]. We only note that the importance of spindle convexity lies, at least partly, in the role intersections of congruent balls play in the study of, for example, the Kneser–Poulsen conjecture, diametrically complete bodies and randomized isoperimetric inequalities for more on this topic and references we suggest to see [2, 8, 11, 12, 17, 19].

If one selectsnindependent and identically distributed random pointsx₁, . . . ,x_nfrom an r-spindle convex discKaccording to the uniform probability distribution, then the intersection of all radiusr discs that containx1, . . . ,xn is a random disc polygonK_(n)^r of radiusrinK. Due to ther-spindle convexity, this random disc polygon is contained inK. In a recent paper, Fodoret al.[10] proved asymptotic formulas for the expectation of the number of vertices, missed area, and perimeter difference ofK_(n)^r under suitable smoothness assumption on the boundary ofK. These asymptotic formulas are generalizations of the corresponding classical results of Rényi and Sulanke in the limit asr→ ∞. Asymptotic estimates on the variance of the number of vertices and missed area were established in [13] for smoothr-spindle convex disc. Ther-spindle convex probability model was generalized tod dimensions in [9] where an asymptotic formula was proved for the expected number of proper facets of the resulting random ball-polytope [9, Theorem 1.1] in the case when a ball of radiusris approximated by random ball-polytopes of radiusr, and asymptotic upper and lower bounds were established for the expected number of proper facets for general convex bodies with sufficiently smooth boundary and suitable radiusr.

The notion of spindle convexity can further be generalized by replacing the radiusrcircular disc by a fixed convex discL. This leads to the notions ofL-convexity andL-spindle convexity, as introduced in [16]. For a historical overview of this topic and references, see the introduction of [16].

LetKandLbe convex discs (to avoid technical complications we always assume that the sets involved are compact). We say thatK isL-convex [16, Definition 1.1] if it is equal to the intersection of all translates ofLthat containK. Of course, ifK isL-convex, then it is also convex in the usual sense. LetX ⊂R² be a set contained in a translate ofL. We denote the intersection of all translates ofLthat containX by [X]_L. The set [X]_Lis called theL-convex hull of X. If L is strictly convex andX has at least two points, then the interior of [X]L

is non-empty.

We say that the convex discK isL-spindle convex [16, Definition 1.2] if it is contained in a translate ofL and for anyx,y∈K it holds that [x,y]L ⊂K. It is clear that ifK is L- convex, then it is also L-spindle convex. The converse is also true (in the plane), see [16, Corollary 3.13, p. 51]. Thus, in our case the two notions of convexity determined byLare

equivalent and can be used interchangeably. We note thatL-convexity andL-spindle convexity can be defined analogously inddimension as well, but fromd3 the two properties are no longer equivalent, see [16, Theorem 3, p. 48].

In classical convexity, a closed convex set is known to have a supporting hyperplane through any of its boundary points. We say that a convex set is smooth if this supporting hyperplane is unique at each boundary point. A similar property holds forL-convex discs too, see [16, Theorem 4, p. 50]: IfKisL-convex,x∈bdKandlis a supporting line ofK throughx, then there exists a translateL+psuch thatx∈L+p,K ⊂L+pandl supportsL+patx. In this case, we callL+pa supporting disc ofKatx. It clearly follows that if bothK andLare smooth, thenKhas a unique supporting disc at each boundary point.

We note that the existence of a supporting translate ofLat each point of bdKis also known as the property thatK slides freely inL, see [27, p. 156].

The L-convex property is invariant under translations ofK, and also under homotheties with ratio strictly between 0 and 1, that is,K isL-convex if and only ifλK +pisL-convex for allλ∈(0,1)andp∈R², see [16, Corollary 3.7, p. 47]).

We study the following probability model. LetK andLbe convex discs withC₊² smooth boundary (twice continuously differentiable with strictly positive curvature everywhere) such thatKisL-convex. TheC₊²property yields that bothKandLare strictly convex. Letx1, . . . ,xn

be independent and identically distributed random points fromK selected according to the uniform probability distribution. We callK_(n)=[x₁, . . . ,x_n]_L a uniform randomL-polygon contained inK. A pointxij ∈ {x1, . . . ,xn}is a vertex ofK₍n) if it is a non-smooth point of bdK_(n). The verticesx_i1, . . .x_ik, 2knofK_(n) divide bdK_(n)into karcs, which we call sides, each of which is a connected arc of the boundary of a translate ofL. Let f₀(K_(n)denote the number of vertices ofK₍n), andA(K \K₍n))the missed area. We investigate the asymptotic behaviour of the expectations of f₀(K_(n)andA(K\K_(n)).

Our paper contains the discussion of two special cases of this probability model. In the first one, we make the following further assumption on the curvatures of the boundaries ofK andL:

xmax∈bdLκL(x) <1< min

y∈bdKκK(y), (1.1)

whereκL(x)is the curvature of bdLatxandκK(y)is the curvature of bdK aty.

We note that [27, Theorem 3.2.12, p. 164] states that for twoC₊² smooth convex discs,K slides freely inLif and only if the curvature of bdK is at least as large as the curvature of bdLin points where the outer unit normals are equal. This condition is clearly satisfied under the assumption (1.1), thus, in this caseKslides freely inLand soK isL-convex.

Under the assumption (1.1), the expected number of vertices and the missed area both behave in a similar manner as in the usual convex case.

THEOREM1.1. With the above assumptions

n→∞lim E(f0(K₍n)))n⁻¹³ = ³

2 3A(K)

5 3 _S¹

(κK(u)−κL(u))¹³

κK(u) du. (1.2) Efron’s identity [5], which, in two dimensions, relates the expectation of the number of vertices and the missed area, can be easily extended to theL-convex probability model as follows

E(f0(K₍n)))=n 1

A(K)E(A(K\K₍n−1))).

Thus we obtain the following corollary of Theorem1.1:

COROLLARY1.1. With the same conditions as above

n→∞lim E(A(K \K_(n)))n²³ = ³

2A²(K)

3

5 3 _S¹

(κK(u)−κL(u))¹³

κK(u) du. (1.3) We note that Theorem1.1and Corollary1.1are contained in Papvári’s Bachelor’s thesis [20]. The proof of Theorem1.1is also from [20].

Note that in both Theorem1.1and Corollary1.1, we get back the corresponding statements of Fodoret al.[10, Theorem 1.1] whenLis a circle of radiusr>1.

In the other special case of the probability model, we investigate in this paper we assume thatK =L, and denote the corresponding randomL-convex polygon byL_(n). This leads to an interesting phenomenon that cannot be observed in the usual convex case. Namely, the expectation of the number of vertices tends to a finite limit determined by onlyL. This has already been pointed out in the case when L=B² in the paper by Fodor et al., see [10, Theorem 1.3], and inddimensions forL=B^d by Fodor [9, Theorem 1.1].

To formulate a precise statement we introduce the following notation. For a unit vector u∈S¹, we denote byw_L(u)=w(u)the distance between the two supporting linesl₁(u)and l2(u)ofLthat are parallel tou. This is the well-known width ofLin the directionu^⊥orthogonal tou.

THEOREM1.2. Let L be a convex disc with C₊² boundary. Then

nlim→∞E(f0(L₍n)))=π

S¹

1

κ_L²(u)·w²(u)du, (1.4)

n→∞lim E(A(L\L_(n)))·n=A(L)π

S¹

1

κL²(u)·w²(u)du. (1.5)

We remark that (1.5) follows from (1.4) by Efron’s identity, thus we focus only on the number of vertices. We note that Theorem1.2is particularly interesting in the case when Lis a convex disc of constant width 1. (The expected number of vertices is clearly scaling invariant.) It is well known that ifLhas constant width 1, thenκ_L⁻¹(u)+κ_L⁻¹(−u)=1 (see, for example, in a more general setting in [11, p. 341]). This impliesκ_L⁻¹(u) <1, and thus

n→∞lim E(f₀(L_(n)))=π

S¹

1

κL²(u)·w²(u)du=π

S¹

1

κL²(u)du<2π². Also, by the arithmetic mean/quadratic mean inequality

1 4 =

κL⁻¹(u)+κL⁻¹(−u) 2

²

κL⁻²(u)+κL⁻²(−u)

2 ,

which implies

n→∞lim E(f₀(L_(n)))=π

S¹

1

κ_L²(u)·w²(u)du=π

S¹

1

κ_L²(u)du π² 2 .

We note that both inequalities are sharp. The upper bound can be approximated by smoothed Reuleaux-polygons. Reuleaux-polygons are not smooth, however with a slight modification

at the vertices one can construct a smooth convex disc of constant width 1 such that the limit of the expectation of the number of the vertices is arbitrarily close to 2π². The lower bound is achieved whenLis a circle, as it was shown in [10, Theorem 1.3].

We also note that ifLis a convex disc withC₊² boundary (but not necessarily of constant width), then the limit is still clearly bounded from below by 2, but one can construct a sausage-like domain with arbitrarily large limit.

§2. Caps of L-convex discs. In this section we assume that (1.1) holds forK andL. We call a subsetC ofK anL-capifC =cl(K\(L+p))for some p∈R². Here cl(·)denotes the closure of a set. Due to the condition (1.1) on the curvatures of the boundaries ofK and L, the curves bdKand bdL+phave exactly two intersection points. These two intersection points divide bdCinto two parts, one belongs to bdK and the other one to bdL+p. Below we state three technical lemmas that will be used in the subsequent arguments. We note that these lemmas are theL-convex analogues of the correspondingr-spindle convex statements in [10], see Lemmas4.1–4.2.

For a smooth convex discM, the unique outer unit normal atx∈bdMis denoted byu(M,x). IfMis also strictly convex, then for eachu∈S¹there exists a unique pointx=x(M,u)such that the outer unit normal of bdMatxisu, that is, the functionsx(M,u)andu(M,x)are inverse to each other. If bdMisC₊², then, with a slight abuse of notation, we useκM(u)=κM(x(M,u)) for the curvature of bdM.

LEMMA2.1. Let K and L be as above. For an L-cap C=cl(K\(L+p)), there exists a unique point x₀∈bdC∩bdK and t 0such that y₀ =x₀−t u(K,x₀)∈bdC∩(bdL+p) and u(L+p,y0)=u(K,x0).

Proof. We may assume, without loss of generality, that p=0. The existence ofx₀ andt follows from the following standard continuity argument. Lety1andy2be the two intersection points of bdK and bdLin the positive direction on bdL. For a pointy∈bdC∩bdL, there exists a uniquet 0 such thatx=y+t u(L,y)∈bdC∩bdK, that is,xis the intersection point of the ray with end pointy, directionu(L,y) and bdK. It is clear that x is strictly monotonically increasing withy. Letϕ=ϕ(y)be the signed angleu(L,y)andu(K,x). Then ϕ(y₁) <0 andϕ(y₂) >0, andϕis a continuous, in fact, continuously differentiable function ofy. Therefore, there is ay₀ such thatϕ(y₀)=0. Thus,y₀ and the correspondingx₀ andt₀ satisfy the statement of the lemma.

Next, we prove the uniqueness ofy₀. On the contrary, assume that there is another point, sayy₀ with the same property.

Letx0andx₀be the point on bdC∩bdK corresponding toy0 andy₀. First, note that d(y₀,y₀) <d(x₀,x₀). (2.1) Clearly, due to theC₊² property of bdK and bdL, the linesx₀y₀andx₀y₀intersect in a point, say p. Let ψ denote the angle ofu(L,y0)andu(L,y₀)(which is the same as the angle of u(K,x₀)andu(K,x₀)).

Due to the relative position ofKandL,d(x₀,p) >d(y₀,p)andd(x₀,p) >d(y₀,p), and all angles of the trianglespy0y₀andpx0x₀are non-obtuse (since the perpendiculars atx0,x₀, and aty₀,y₀, are supporting lines ofK, andL, respectively). If∠py₀y₀<∠px₀x₀, then lety₀be the intersection point ofpy₀and the line parallel toy₀y₀throughx₀. Then, since∠py₀y₀ >∠px₀x₀, it holds that d(y₀,y0) <d(x₀,y₀) <d(x₀,x0), proving (2.1) in this case. The case when

∠py₀y0>∠px₀x0 is similar.

Again, by the conditions on the curvatures of bdK and bdL, the shorter open arc of the unit circle with end pointsy0andy₀is inK but outside ofL. Therefore, its lengthhis larger than the lengthsof the arc of bdLfromy₀toy₀. Similarly, the shorter closed unit circular arc with end pointsx₀andx₀is completely inK, and thus its lengthhis less than the length sof the arc of bdK fromx0tox₀. It follows from (2.1) thath<h. In summary,

s<h<h< s.

On the other hand, ifIK denotes the part of bdK betweenx0 andx₀, andIL denotes the part of bdLbetweeny₀andy₀, then it follows from the conditions on the curvatures of bdLand bdKthat

s=

IL

ds>

IL

κL(s)ds=ψ=

IK

κK(s)ds>

IK

ds=s,

which is a contradiction.

This (unique) pointx₀ =x(K,u)is usually called thevertexofC and the correspondingt is theheight. Since theL-capCis uniquely determined by its vertex and height, we introduce the notationC(u,t)to denote such a cap. This, in fact, provides a parametrization ofL-caps in terms of a unit vector and a (sufficiently small) positive real number. LetA(u,t)=A(C(u,t)), and let (u,t)be the arc-length ofC∩(bdL+p).

LEMMA2.2. Let K and L be as above. Then, for a fixed u∈S¹, the following hold:

t→0lim⁺ (u,t)·t⁻¹² =2·

2

κK(u)−κL(u), (2.2)

t→0lim⁺A(u,t)·t⁻³² =4 3·

2

κK(u)−κL(u). (2.3)

The proof of Lemma2.2is very similar to that in [10, Lemma 4.2, p. 906], thus we omit the details. The main idea of the argument is that we assume thatx=(0,0)andu=(0,−1). Then, in a sufficiently small open neighbourhood of the origin, bdK is the graph of aC² smooth convex function f(x). Then we use the second-order Taylor expansion of f around the origin from which we obtain the statements of the lemma by simple integration.

For two points,x,y∈K there are the two (unique) translates ofLsuch that each translate contains bothxandyon its boundary. We denote theL-caps determined by these translates ofL byC₋(x,y)andC₊(x,y)with the assumption thatA₋(x,y)=A(C₋(x,y))A(C₊(x,y))= A₊(x,y).

LEMMA 2.3. Let K and L be as above. Then there exists a constantδ >0such that for any x₁,x₂ ∈K, it holds that A₊(x₁,x₂) > δ. The constantδdepends only on K and L.

We also omit the proof of Lemma2.3as it is essentially the same as that in [10, Lemma, 4.3, p. 906]; it uses only the conditions on the curvatures of bdL and bdK and a simple compactness argument.

Finally, we need the existence of a rolling circle inK. We say that a circle of radius >0 rolls freely inKif eachx∈bdKis in a closed circular disc of radiusthat is fully contained inK. It follows from Blaschke’s result [3] that if bdKisC²smooth with the above conditions

on its curvature, then there exists a circle of radius 0< <1 (cf. [10, p. 906] and [14]) that rolls freely inK. Thus, according to (2.3), there exists a 0<t^∗< , such that for allu∈S¹

A(u,t) 1 2

4 3

2 1/−κ^∗

t³², ift ∈[0,t^∗], (2.4) whereκ^∗=min_u∈S¹κL(u).

§3. Proof of the Theorem1.1. Our argument is essentially based on ideas that originated from Rényi and Sulanke [22], and which were also used in the spindle convex setting in [10].

Here we generalize and apply them to theL-convex probability model.

A pair of random pointsxi,xjforms an edge ofK₍n)if at least one of theL-capsC₋(xi,xj)and C₊(x_i,x_j)contains no other points ofx₁, . . . ,x_n. LetA₋(x,y)=A(C₋(x,y))andA₊(x,y)= A(C₊(x,y)). Then

E f₀(K_(n))

= 1 A(K)²

n

2 _K

K

1− A₋(x₁,x₂) A(K)

_n−2

+

1−A₊(x₁,x₂) A(K)

_n−2

dx₁dx₂, (3.1)

where integration is with respect to the Lebesgue measure inR².

Using a similar argument to the one in [10, p. 907] one can show that the contribution of the second term of (3.1) in the limit asn→ ∞is negligible, in fact, is exponentially small.

For the sake of completeness, we give a detailed proof. For any fixedα∈R, it follows from Lemma2.3that

n→∞lim n^α 1 A(K)²

n

2 _K

K

1−A₊(x1,x2) A(K)

n−2

dx1dx2

lim

n→∞n^α 1 A(K)²

n

2 _K

K

1− δ

A(K) n−2

dx1dx2

lim

n→∞n^α 1 A(K)²

n

2 _K

K

e^{−δ(n−2)A(K)} dx1dx2

= lim

n→∞n^α n

2

e^{−δ(n−2)A(K)}

=0.

Note that the same argument shows that the contribution of those pairs x1,x2 for which A₋(x₁,x₂) > δis also negligible in the limit. Thus,

nlim→∞E f0(K₍n)) n⁻¹³

= lim

n→∞n⁻¹³ 1 A(K)²

n

2 _K

K

1− A₋(x₁,x₂) A(K)

_n−2

1(A₋(x₁,x₂) < δ)dx₁dx₂, (3.2) where1(·)denotes the indicator function of an event. In the rest of the proof, we evaluate the right-hand side of (3.2).

Define a function:⊂(S¹,R,S¹,S¹)→K×K by

(u,t,u₁,u₂)=(x₁,x₂), (3.3) where u∈S¹ and t t₀ are such that C(u,t)=C₋(x₁,x₂). Let L(u,t) denote the arc C(u,t)∩(bdL+x(K,u)−x(L,u)−t u). Thusx1,x2∈L(u,t). The outer unit normals of L+x(K,u)−x(L,u)−t uon the arc L(u,t)determine a connected arc of S¹, which we denote byL^∗(u,t). Letu₁,u₂ be the outer unit normals ofL+x(K,u)−x(L,u)−t uatx₁ andx2. Thus,

x_i =x(K,u)−x(L,u)−t u+x(L,u_i), i=1,2, (3.4) withu₁,u₂ ∈L^∗(u,t).

Lemma2.1guarantees the uniqueness of the vertex and height of anL-cap, thusis well defined, bijective and differentiable (see the Appendix) on a suitable domain of(u,t,u1,u2) with the possible exception of a set of measure zero. The Jacobian of the transformationis

|J| = |u₁×u₂| κL(u₁)κL(u₂)

1

κL(u)− 1 κK(u)+t

, (3.5)

see the details in the Appendix. From (3.2) and (3.5), we obtain

n→∞lim E(f0(K₍n)))n⁻¹³

= lim

n→∞n⁻¹³ 1 A(K)²

n 2 _S1

t^∗(u) 0

L^∗(u,t)

L^∗(u,t)

1−A(u,t) A(K)

_n−2

× |u₁×u₂| κL(u₁)κL(u₂)

1

κL(u)− 1 κK(u)+t

du₁du₂dt du, (3.6) with a suitablet^∗(u)depending only onKandL.

We note that in (3.6) we can replacet^∗(u)by any fixed 0<t₁ t^∗(u)and the limit remains unchanged. We choose a suitable 0<t1 t^∗(u)such thatA(u,t)δfor allt1t t^∗(u) and allu∈S¹.

Now we split the domain of integration with respect to t into two parts. Let h(n)= (clnn/n)^2/3, wherecis a suitable positive constant specified below in the proof. There exists n₀∈N, such that ifn>n₀, thenh(n) <t₁. Furthermore, there also existsγ1>0 constant such thatA(u,t) > γ1h(n)^3/2for allu∈S¹andh(n) <t t₁. Foru∈S¹and 0t t₁, let

I^∗(u,t)=

L^∗(u,t)

L^∗(u,t)

|u₁×u₂|

κL(u₁)κL(u₂)du₁du₂ and

k(u,t)= 1

κL(u)− 1 κK(u)+t. LEMMA3.1. Let h(n)be defined as above. Then

n→∞lim n⁻¹³ 1 A(K)²

n 2 _S1

_t1

h(n)

1− A(u,t) A(K)

_n−2

k(u,t)I^∗(u,t)dt du=0. Proof. Note that there exists a universal constantγ2>0 such that

k(u,t)I^∗(u,t)γ2

for allu∈S¹ and 0<t t₁. Hence, for a fixedu∈S¹and anyn>n₀, it holds t1

h(n)

1− A(u,t) A(K)

_n−2

k(u,t)I^∗(u,t)dt

_t1

h(n)

1−A(u,t) A(K)

n−2

dt

_t1

h(n)

1− γ1h(n)³² A(K)

n−2

dt

t1

0

1− γ1c(lnn/n) A(K)

_n−2 dt n^−A(Kγ1c).

Ifc>5A(K)/(3γ1), then n⁻¹³ 1

A(K)² n

2 _S1

t1

h(n)

1− A(u,t) A(K)

_n−2

k(u,t)I^∗(u,t)dt du n⁵³n^−A(K)γ1c ,

which clearly converges to 0 asn→ ∞.

Letε >0 be fixed. There exists a 0<t_ε <t₁ such that for all 0<t <t_ε,u∈S¹and for anyu₁,u₂∈L^∗(u,t)

(1−ε) 1

κ_L²(u) < 1

κL(u₁)κL(u₂) < (1+ε) 1 κ_L²(u). Then, with the notation

I(u,t)=

L^∗(u,t)

L^∗(u,t)|u1×u2|du1du2, we obtain

1−ε

κ_L²(u)I(u,t) <I^∗(u,t) < 1+ε κ_L²(u)I(u,t).

And

I(u,t)=2( ^∗(u,t)−sin ^∗(u,t)), where ^∗(u,t)is the length of the arcL^∗(u,t)⊂S¹. Thus,

I^∗(u,t)=(1+O(ε))2( ^∗(u,t)−sin ^∗(u,t))

κ_L²(u) . (3.7)

Integrating in (3.6) with respect tou₁andu₂, and using Lemma3.1, we obtain

n→∞lim E(f0(K_(n)))n⁻¹³ =(1+O(ε)) lim

n→∞n⁻¹³ 2 A(K)²

n 2

×

S¹

_h(n)

0

1−A(u,t) A(K)

n−2

k(u,t) ^∗(u,t)−sin ^∗(u,t) κ_L²(u) dt du.

Letn₁be such that 0<h(n) <t_εifn>n₁. Now assume thatn>max{n₀,n₁}and define θn(u)=n⁻¹³

n 2

h(n) 0

1−A(u,t) A(K)

_n−2

k(u,t) ^∗(u,t)−sin ^∗(u,t)

κ_L²(u) dt. (3.8) Then

n→∞lim E(f₀(K_(n)))n⁻¹³ =(1+O(ε)) lim

n→∞

2 A(K)²

S¹

θn(u)du. (3.9) We recall from [4, (11), p. 2290] that, for anyβ0, ω >0 andα >0 it holds that

_g(n)

0

t^β(1−ωt^α)ⁿdt ∼ 1 αω^β+1α

β+1 α

n^−β+1α , (3.10)

asn→ ∞, assuming that

(β+α+1)lnn αωn

¹_α

<g(n) < ω^−1α for sufficiently largen.

In order to use Lebesgue’s dominated convergence theorem for (3.9), we need to show that the functionsθn(u)are uniformly bounded onS¹. Clearly, there exists aγ3>0 constant such that for all 0<t <t_εandu∈S¹we have

∗(u,t)−sin ^∗(u,t)

κ_L²(u) < γ3t³², and

k(u,t)= 1

κL(u)− 1

κK(u)+t < γ4, for a suitableγ4>0 constant. From (3.10) and (3.8) with

α= 3

2, β = 3 2, ω=

2 3

2 1/−κ^∗

A(K) , (3.11)

whereκ^∗is the minimum ofκL(u)foru∈S¹, andis the radius of the rolling circle ofKas in (2.4), it follows that there existsγ5>0 such thatθn(u) < γ5for allu∈S¹and sufficiently largen. Thus, by Lebesgue’s dominated convergence theorem

n→∞lim E(f₀(K_(n)))n⁻¹³ =(1+O(ε)) 2 A(K)²

S¹

n→∞lim θn(u)du. (3.12) Now assume thatt_ε>0 is so small, that the following two conditions also hold for all 0<t <t_εandu∈S¹

(1−ε)4 3

2

κK(u)−κL(u)t³² <A(u,t) < (1+ε)4 3

2

κK(u)−κL(u)t³², (3.13) and

(1−ε)4 3

2 κK(u)−κL(u)

³₂

t³² < ³(u,t)

6 < (1+ε)4 3

2 κK(u)−κL(u)

³₂ t³², as a result of Lemma2.2. Using the Taylor series expansion of sinxaround 0, and the fact that lim_t→0^{+ ∗}(u,t)/ (u,t)=κL(u), we obtain that for a sufficiently smallt_ε>0 it holds

that

∗(u,t)−sin ^∗(u,t)

κL²(u) = ( ^∗(u,t))³

6κL²(u) +O ( ^∗(u,t))⁵

=(1+O(ε))κL(u) ³(u,t)

6 .

Thus, we obtain

∗(u,t)−sin ^∗(u,t)

κ_L²(u) =(1+O(ε))κL(u)4 3

2 κK(u)−κL(u)

³₂

t³². (3.14) From (3.13) and (3.14), it follows that

n→∞lim E(f₀(K_(n)))n⁻¹³

=(1+O(ε)) 2 A(K)²

S¹

n→∞lim n⁻¹³ n

2

h(n) 0

⎛

⎝1−

4 3

2 κK(u)−κL(u)

A(K) t³²

⎞

⎠

n−2

×k(u,t)κL(u)4 3

2 κK(u)−κL(u)

³₂

t³²dt du. Therefore,

nlim→∞E(f0(K₍n)))n⁻¹³ =(1+O(ε)) 4 3A(K)²

S¹

2 κK(u)−κL(u)

³₂ κL(u)

× lim

n→∞n⁵³ h(n)

0

t³²

⎛

⎝1−

4 3

2 κK(u)−κL(u)

A(K) t³²

⎞

⎠

n−2

k(u,t)dt du.

Now, the substitutionα=β =3/2 andω=(4√

2/(3A(K)))(κK(u)−κL(u))^−1/2 yields

n→∞lim E(f0(K_(n)))n⁻¹³ =(1+O(ε)) 4 3A(K)²

S¹

2 κK(u)−κL(u)

³₂ κL(u)

×

nlim→∞n⁵³ _h(n)

0

t^β(1−ωt^α)ⁿ⁻² 1

κL(u)− 1 κK(u)

dt (3.15)

+ lim

n→∞n⁵³ h(n)

0

t^β+1(1−ωt^α)ⁿ⁻²dt

du. (3.16)

It follows from the asymptotic formula (3.10) that the term (3.16) is 0. Applying (3.10) to (3.15), we obtain

n→∞lim E(f₀(K_(n)))n⁻¹³ =(1+O(ε)) 2⁷² 3A(K)²

S¹

1 κK(u)−κL(u)

³₂

×κK(u)−κL(u) κK(u) n⁵³2

3

⎛

⎝

4 3

2 κK(u)−κL(u)

A(K)

⎞

⎠

−⁵₃

5

3

n⁻⁵³du.

After simplification, we get

n→∞lim E(f₀(K_(n)))n⁻¹³ =(1+O(ε))³

2 3A(K)

5 3 _S¹

(κK(u)−κL(u))¹³ κK(u) du. Sinceε >0 was arbitrary, this completes the proof of Theorem1.1.

Remark3.1. IfK ⊂R²is a convex disc, then we denote integration on bdK with respect to the arc-length by

bdK. . .dx. It is well known that if bdK isC₊², then for any measurable function f(u)onS¹ it holds that

S¹

f(u)du=

bdK

f(u(K,x))κK(x)dx, (3.17) see, for example, [27, (2.5.30)]. With the help of (3.17), the statement of Theorem1.1can also be phrased slightly differently in the form

n→∞lim E(f0(K_(n)))n⁻¹³ = ³

2 3A(K)

5

3 _bdK(κK(x)−κL(x))¹³dx.

§4. The K =L case. In this section, we investigate the case whenK =Land thus turn to the proof of Theorem 1.2. This is a direct generalization of [10, Theorem 1.3]. Since the argument closely follows the proof of Theorem1.1in §3, we only point out the major differences in the calculations. First we note that the analogue of Lemma2.1is true in the caseK =Las well. A general cap is of the formC=cl(L\(L+p)), and it clearly follows that the vertexx0of the cap is the unique point in bdLwhere the unit outer normal is−p/|p|, while the height ist = |p|. We are going to use the notionC(u,t),A(u,t), etc. as before with the assumption thatK =L. Next we need a variant of Lemma2.2.

LEMMA4.1. Let L be a convex disc with C₊² boundary. Then

tlim→0⁺

∗(u,t)=π, (4.1)

t→0lim⁺A(u,t)·t⁻¹ =w(u). (4.2) The proof of the Lemma4.1is simple, as the intersection points of bdLand bdL−t utend to the points of tangency of the supporting linesl₁(u)andl₂(u)ast →0+.

We use the reparametrizationas introduced in (3.3), and have that

|J| = |u₁×u₂| κL(u₁)κL(u₂)t. After the integral transformation, we obtain

n→∞lim E(f₀(L_(n)))= lim

n→∞

1 A(L)²

n 2 _S¹

_t^∗_(u)

0

L^∗(u,t)

L^∗(u,t)

1−A(u,t) A(L)

n−2

× |u1×u2|

κL(u1)κL(u2)t du1du2dt du.

In the next step, as in Lemma 3.1, we split the domain of integration in t. From (4.2), it follows that there is a universal constantcˆ= ˆc(L)such that A(u,t) >ctˆ. We set