GUO Qi

（School of Mathematics and Physics，SUST，Suzhou 215009，China）

The well-known Reuleaux polygons are a class of very important convex domains in the Eclidean planeR2.Inspired by these polygons，the convex bodies of constant width were introduced and studied in higher dimensional spaces.Reuleaux polygons and the general convex bodies of constant width have many extremal properties（see[1-6]and the references therein）.

The support function of convex sets is a kind of useful tool in studying convex sets.Several geometric invariants of convex bodies，such as some measures of asymmetry，the Banach-Mazur/Haudorff distances and the mixed volumes etc.，can be defined or formulated through support functions （see[7-9]and the references therein）.However，it is a little strange that there seems no explicit expressions of the support function of Reuleaux polygons available in literature.

So，in this short note，for the Reuleaux polygons，we present an explicit expression of their support functions in terms of inner product and a concrete calculating formula of their p-measures of asymmetry[8]（1≤p<+∞） in terms of the usual Riemann integral.The results obtained and the method adopted in the article should be useful in the further studies.

1 Preliminaries

Rndenotes the Euclidean n-space with the usual inner product 〈·，·〉 and the induced norm||·||.For nonempty C?Rn，coC，coneC denote the convex hull and the conical hull of C respectively.It is known that if C is compact，then so is coC （see[10]）.

For compact convex set C?Rnand u∈Rn，denote F（C，u）:={x∈C|〈u，x〉=maxy∈C〈u，y〉}，called the support set of C in the direction u.For any ω?Sn-1，the unit sphere ofRn，we define，where H denotes the Hausdorff measure which coincides with the （（n-1）-dimensional） Lebesgue measure on the family of measurable ω’s.Sn－1（C，·） is called the surface area measure of C（see[7]）.

Given a bounded C?Rn，its support function h（C，·）:Rn→Ris defined by

It is easy to check that for any bounded C?Rn，h（C，·） is sublinear and that h（C，·）=h（coC，·）（see[7]）.Since h（C，·） is positively homogenous，one may consider，as we often do in this paper，the support functions restricted onSn-1only.

More generally，we define，for any z∈Rn（see[8]）called the support function of C with respect to z.Clearly，ho（C，·）=h（C，·），where “o” denotes the origin ofRn.The relation between h（C，·） and hz（C，·） with general z is shown in the following lemma，which will be used later.

Lemma 1For any bounded C?Rnand z∈R，we have

Proof.For u∈S1，〉.The proof is completed. □

The following simple fact，stated as a lemma，will also be used later.

Lemma 2[7]h（C，·）=〈z，·〉 for some z∈Rnif and only if C is the singleton{z}.

Given a compact convex C?Rnwith nonempty interior and a fixed z∈int（C） （the interior of C），we define a probability measure mz（C，·） onSn-1by，for any measurable ω?Sn-1,

Then write，

where

Definition 1[8]For compact convex C?Rnwith nonempty interior，we define its p-measure of asymmetry（1≤p≤∞）asp（C） by

A point z∈int（C）satisfying that μp（C，z）=asp（C） is called a p-critical point of C.

Remark 1It is proved in[8]that for 1

In this note，we often use Im:={1，2，…，m}as index set and we adopt the congruence addition （denoted still by the symbol“+”） with modulus m as the group composition on Im.Thus，we have always k+m=k for all k∈Im.

A convex set Pm?R2is called an m-polygon if Pm=co{v1，v2，…，vm}for some v1，v2，…，vm∈R2and each viis a vertex of P.For convenience，in this paper we always assume that v1，v2，…，vmis indexed in succession anticlockwisely.An m-polygon is called to be regular if all vilocate at a circle with center at some point v*and all||vi-vi+1||are equal for i∈Im.In such a case，v*is exactly the centroid of P.

Let Pmbe a regular m-polygon with vertices v1，v2，…，vm，where m≥3 is odd.For each i∈Im，it is easy to see that||vi-vi-（m-1）/2||=||vi-vi+（m+1）/2||and that all||vi-vi-（m-1）/2||are equal by the regularity （drawing a picture to check）.We denote w:=||vi-vi-（m-1）/2||and then denote by B（vi，w） the closed ball with radius w and centered at vi（observing that vi-（m-1）/2and vi+（m+1）/2are in the boundary of B（vi，w））.Then a regular Reuleaux m-polygon Rm（of width w，with centroid at v*） is defined as，which is a compact convex domain of constant width w.R3is just the well-known Reuleaux triangle.

In the rest of this paper，we always assume （unless mentioned otherwise） that Pm（and so Rm） is regular and has its centroid at o.We mention also that there are also non-regular Reuleaux m-polygons for odd m>3，which are constructed in a similar way.

Given a regular m-polygon P=co{v1，v2，…，vm} （m≥3），we denote

each of which is a convex cone.Observe that

The following simple fact，stated as a lemma as well，can be checked easily，so we leave its proof to the readers.

Lemma 3For any regular m-polygon P（m≥3），we have

2 The support functions of Reuleaux polygons

In this section，we present an explicit expression of support function for the Reuleaux polygons.

Theorem 1Let Pm=co{v1，v2，…，vm}be a regular m-polygon （m≥3，odd）with its centroid at o.Then

where w=||vi-vi-（m-1）/2||for any i∈Im.

Proof.For u∈R2，by Lemma 3，u∈Ki+for some i∈Imor u∈Kj-for some j∈Im.

If u∈Ki+，then by Lemma 1 and the construction of Rm，we have

If u∈Kj-，then by the construction of Rmand Lemma 2，we have

The proof is completed.□

Letand.Then Δ:=co{v1，v2，v3}is a regular triangle with its centroid at the origin o，and so R（Δ）is a regular Reuleaux triangle of width 1（observing that the length of each edge of Δ is 1） with its centroid at o.It is easy to check that

K1+（Δ）=co{（cosπ/2，sinπ/2），（cos5π/6，sin5π/6）}；K1－（Δ）=intco{（cos3π/2，sin3π/2），（cos11π/6，sin11π/6）}K2+（Δ）=co{（cos7π/6，sin7π/6），（cos3π/2，sin3π/2）}；K2－（Δ）=intco{（cosπ/6，sinπ/6），（cosπ/2，sinπ/2）}

K3+（Δ）=co{（cos11π/6，sin11π/6，（cosπ/6，sinπ/6）}；K3－（Δ）=intco{（cos5π/6，sin5π/6），（cos7π/6，sin7π/6）}Therefore，the support function，restricted onS1，of R（Δ） can be written explicitly as

As an application of Theorem 1，we present a precise expression of support function of a Reuleaux triangle.

where u=（cosθ，sinθ）∈S1，and we write[11π/6，13π/6]instead of[0，π/6]∪[11π/6，2π].

3 The p-measures of asymmetry of Reuleaux polygons

In this section，we present a calculating formula of the p-measures of asymmetry for Reuleaux m-polygons.

In order to calculate the p-measures of asymmetry for Rm，we observe first that，since for，it follows that S1（Rm，ω）=0 for any.Then，we observe that，since，it follows that S1（Rm，ω）=ws（ω） for any，where s（·） denotes the usual arc length measure on the unit circleS1.

Theorem 2Let Rmbe a regular Reureaux triangle.Then，for 1≤p<+∞，

In particular

where

Remark 2When p=1 amd m=3，we have.So，we obtain the 1-measure of asymmetry of the Reuleaux triangle

an interesting number involving only 3 （the number of edges of triangle） and π（related to the circle）,and much less than its Minkowski measure of asymmetry

Proof.Denoteand.Since the p-measures are affinely invariant，without loss of generality，we may assume that v1=（－a，0） （a>0 to be fixed later），（v1+（m-1）/2－v1）｜｜e′，（v1+（m+1）/2－v1）｜｜e″（so K1+=cone{（e′，e″）}），w=1 and that its centroid is o.Thus，noticing that the p-critical point（1≤p<+∞） of Rmis o clearly by symmetry （recalling that its p-critical point is unique for 1

Now，we are going to fix a.First，we know that the length of each edge of Pmisand that each inner angle of Pmis.Then，calculating on the right triangle co{o，v1+（m+1）/2，，we have

Next，denote bythe circular-sector-like domain determined by 0，v1+（m-1）/2，v1+（m+1）/2and bythe circular sector determined by v1，v1+（m-1）/2，v1+（m+1）/2.Then，denote Δ1:=co{v1，v1+（m-1）/2，v1+（m+1）/2}and Δ2:=co{o，v1+（m-1）/2，v1+（m+1）/2}，which are triangles.Thus，through elementary calculations we obtain

which leads to

Observing，we have

In particular，when p=1，we have

The proof is completed.□

Final RemarkConcrete results obtained in this note are simply an explicit expression of support functions of the Reuleaux polygons and a calculating formula of p-measures of asymmetry for Reuleaux polygons.Although these results are elementary，they might play roles in solving some other problems.Also，the method adopted here for “calculating” support functions is inspiring.Hopefully，one may calculate the support functions of other compact convex domains constructed in the same manner as Reuleaux polygons and of similar convex bodies in higher dimensions.

：

[1]JIN H L，GUO Q.Asymmetry of convex bodies of constant width[J].Discrete Comput Geom，2012，47：415-423.

[2]JIN H L，GUO Q.A note on the extremal bodies of constant width for the Minkowski measure[J].Geom Dedicata，2013，164：227-229.

[3]GUO Q，JIN H L.On a measure of asymmetry for Reuleaux polygons[J].J Geom，2011，102：73-79.

[4]JIN H L，GUO Q.On the asymmetry for convex domains of constant width[J].Commun Math Res，2010，26（2）：176-182

[5]CHARKERIAN G D，GROEMER H.Convex bodies of constant width[C]//Convexity and Its Applications，Basel：Birkh?ser，1983：49-96.

[6]SALLEE G T.Maximal area of Reuleaux polygons[J].Can Math Bull，1970，13：175-179.

[7]SCHNEIDER R.Convex Bodies：The Brunn-Minkowski Theory[M].2th ed.Cambridge：Camb Univ Press，2014.

[8]GUO Q.On p-measures of asymmetry for convex bodies[J].Adv Geom，2012，12：287-301.

[9]TOTH G.Measures of Symmetry for Convex Sets and Stability[M].Universitext：Springer，2015.

[10]HIRIART-URRUTY J B，LEMARéMATIQUES C.Fundamentals of Convex Analysis[M].Berlin Heidelberg：Springer-Verlag，2001.