ON A BRUNN-MINKOWSKI THEOREM FOR A GEOMETRIC DOMAIN FUNCTIONAL CONSIDERED BY AVHADIEV

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Brunn-Minkowski Theorem G. Keady vol. 8, iss. 2, art. 33, 2007

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ON A BRUNN-MINKOWSKI THEOREM FOR A GEOMETRIC DOMAIN FUNCTIONAL

CONSIDERED BY AVHADIEV

G. KEADY

School of Mathematics and Statistics University of Western Australia 6009, Australia

EMail:keady@maths.uwa.edu.au

Received: 28 December, 2006

Accepted: 26 April, 2007

Communicated by: D. Hinton 2000 AMS Sub. Class.: 26D15, 52A40.

Key words: Brunn-Minkowski, Prekopa-Leindler.

Abstract: Suppose two bounded subsets of Rⁿare given. Parametrise the Minkowski combination of these sets byt. The Classical Brunn-Minkowski Theorem asserts that the1/n-th power of the volume of the convex combination is a concave function oft. A Brunn-Minkowski-style theorem is established for another geometric domain functional.

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1. Introduction

LetΩbe a bounded domain inRⁿ. Define

(1.1) I(k, ∂Ω) =

Z

Ω

dist(z, ∂Ω)^kdµ_z fork > 0.

Heredist(z,Ω)denotes the distance of the point z ∈ Ωto the boundary∂Ωof Ω.

The integration uses the ordinary measure inRⁿand is over allz ∈Ω. Whenn = 2 andk = 1 this functional was introduced, in [1], in bounds of the torsional rigidity P(Ω)of plane domainsΩ. See also [10] where the inequalities

(1.2) I(2, ∂Ω)

I(2, ∂B1) ≤ P(Ω)

P(B1) ≤ 128 3

I(2, ∂Ω) I(2, ∂B1) are presented. HereB₁is the unit disk and

I(2, ∂B₁) = π

6 = |B₁|² 6π .

This inequality is one of many relating domain functionals such as these: see [9,2, 7]. As an example, proved in [9], we instance

(1.3) ( ˙r(Ω))⁴ ≤ P(Ω) P(B₁) ≤

|Ω|

|B₁| 2

≤

|∂Ω|

|∂B₁| 4

giving bounds for the torsional rigidity in terms of the inner-mapping radiusr, the˙ area|Ω|and the perimeter|∂Ω|.

We next define the Minkowski sum of domains by

Ω₀+ Ω₁ :={z₀+z₁|z₀ ∈Ω₀, z₁ ∈Ω₁},

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and

Ω(t) :={(1−t)z₀+tz₁|z₀ ∈Ω₀, z₁ ∈Ω₁}, 0≤t≤1.

The classical Brunn-Minkowski Theorem in the plane is thatp

|Ω(t)|is a concave function oftfor0≤t≤1, and it also happens that|∂Ω(t)|is, for convexΩ, a linear, hence concave, function of t. Given a nonnegative quasiconcave function f(t) for which, withα >0,f(t)^αis concave, we say thatfisα-concave. In [3] it was shown that, for convex domainsΩ, the torsional rigidity satisfies a Brunn-Minkowski style theorem: specifically P(Ω(t)) is 1/4-concave. Thus inequalities (1.3) show that the 1/4-concave functionP(Ω(t))is sandwiched between the 1/4-concave functions

|Ω(t)|² and |∂Ω(t)|⁴. In [6] it is shown that the polar moment of inertia I_c(Ω(t)) about the centroid ofΩ, for which

(1.4)

|Ω|

|B₁| 2

≤ I_c(Ω) I_c(B₁) ≤

|∂Ω|

|∂B₁| 4

,

holds, is also 1/4-concave. (The 1/4-concavity ofr(Ω(t))˙ ⁴ has also been established by Borell.) In this paper we show that the same 1/4-concavity of the domain functions holds for the quantities in inequalities (1.2). Our main result will be the fol- lowing.

Theorem 1.1. Let K denote the set of convex domains in Rⁿ. For Ω₀, Ω₁ ∈ K, I(k, ∂Ω(t))is1/(n+k)-concave int.

Our proof is an application of the Prekopa-Leindler inequality, Theorem2.2 be- low.

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2. Proofs

The proof will use two little lemmas, Theorems2.1and2.3, and one major theorem, the Prekopa-Leindler Theorem2.2. None of these three results is new: the new item in this paper is their use.

Theorem 2.1 (Knothe). Let0< t <1andΩ0, Ω1 ∈ K. With z_t= (1−t)z₀+tz₁,

we have

(2.1) dist(z_t, ∂Ω(t))≥(1−t) dist(z₀, ∂Ω₀) +tdist(z₁, ∂Ω₁).

Proof. Letz_t ∈ Ω(t)be as above. Denote the usual Euclidean norm with | · |. Let w_t ∈∂Ω(t)be a point such that

|z_t−w_t|= dist(z_t, ∂Ω(t)).

Define the directionuby

u= zt−wt

|z_t−w_t|.

Define v₀ ∈ Ω₀, and v₁ ∈ Ω₁ as the points on these boundaries which are on the rays, in directionu, fromz₀andz₁respectively. Thus

v₀ =z₀ +|z₀−v₀|u, v₁ =z₁+|z₁−v₁|u.

Now letpbe any unit vector perpendicular tou. The preceding definitions give that hw_t−((1−t)v₀ +tv₁), pi= 0,

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from which, on defining

v_t= (1−t)v₀+tv₁ we havew_t=v_t+ηu.

for some numberη. Now, we do not know (or care) ifv_tis on the boundary ofΩ(t), but we do know that v_t is in the closed set Ω(t). Using the convexity of D(t) we have thatv_tis on the ray joiningz_twithw_t, and betweenz_tandw_t. From this,

dist(z_t, ∂Ω(t)) =|z_t−w_t| ≥ |z_t−v_t|,

= (1−t)|z0−v0|+t|z1−v1|,

≥(1−t) dist(z₀, ∂Ω₀) +tdist(z₁, ∂Ω₁), as required.

Theorem 2.2 (Prekopa-Leindler). Let0< t <1and letf₀,f₁, andhbe nonnega- tive integrable functions onRⁿsatisfying

(2.2) h((1−t)x+ty)≥f₀(x)^1−tf₁(y)^t, for allx, y ∈Rⁿ. Then

(2.3)

Z

Rⁿ

h(x)dx≥ Z

Rⁿ

f₀(x)dx

1−tZ

Rⁿ

f₁(x)dx t

. For references to proofs, see [5].

Theorem 2.3 (Homogeneity Lemma). IfF is positive and homogeneous of degree 1,

F(sΩ) = sF(Ω) ∀s >0,Ω, and quasiconcave

(2.4) F(Ω(t))≥min(F(Ω(0)), F(Ω(1))) ∀0≤t ≤1, ∀Ω0,Ω1 ∈ K,

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then it is concave:

F(Ω(t))≥(1−t)F(Ω(0)) +tF(Ω(1)) ∀0≤t≤1.

Proof. See [5]. ReplaceΩ₀byΩ₀/F(Ω₀),Ω₁byΩ₁/F(Ω₁). Using the homogeneity of degree 1, and applying (2.4), we have

F

(1−t) Ω₀

F(Ω₀) +t Ω₁ F(Ω₁)

≥1. With

t = F(Ω₁)

F(Ω0) +F(Ω1) , so(1−t) = F(Ω₀) F(Ω0) +F(Ω1) , the last inequality onF becomes

F

Ω₀+ Ω₁ F(Ω₀) +F(Ω₁)

≥1.

Finally, using the homogeneity we have

F(Ω0+ Ω1)≥F(Ω0) +F(Ω1), and using homogeneity again,

F((1−t)Ω₀+tΩ₁)≥(1−t)F(Ω₀) +tF(Ω₁), as required.

Proof of the Main Theorem1.1. Knothe’s Lemma2.1and the AGM inequality give (2.5) dist(z_t, ∂Ω(t))≥dist(z₀, ∂Ω₀)^(1−t)dist(z₁, ∂Ω₁)^t,

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and similarly for any positivek-th power of the distance. Denote the characteristic function ofΩbyχ_Ω. A standard argument, as given in [5] for example, establishes that

χ_Ω(t)((1−t)z₀+tz₁)≥χ_Ω₀(z₀)^1−tχ_Ω₁(z₁)^t. So, with

h(z) = dist(z, ∂Ω(t))χ_Ω(t)(z), f₀(z) = dist(z, ∂Ω₀)χ_Ω₀(z), f₁(z) = dist(z, ∂Ω₁)χ_Ω₁(z),

the conditions of the Prekopa-Leindler Theorem are satisfied. This gives thatI(k, ∂Ω(t)) is log-concave in t. Now define F(Ω(t)) := I(k, ∂Ω(t))^1/(n+k). The function F is quasiconcave in t (as it inherits the stronger property of logconcavity in t from I(k, ∂Ω(t))). Since I(k, ∂Ω(t)) is homogeneous of degree n +k, F is homogeneous of degree 1. The Homogeneity Lemma applied toF yields thatI(k, ∂Ω(t))is 1/(n+k)-concave.

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Acknowledgements

I thank Richard Gardner for calling to my attention that I had omitted Borell’s paper [3] from my survey paper [7].

I thank Sever Dragomir for his hospitality during a visit in 2006 to Victoria Uni- versity, and for continuing his interest, begun in [4], in the elastic torsion problem in convex domains.

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References

[1] F.G. AVHADIEV, Solution of generalized St. Venant problem, Matem. Sborn., 189(12) (1998), 3–12. (Russian).

[2] C. BANDLE, Isoperimetric Inequalities and Applications, Pitman, 1980.

[3] C. BORELL, Greenian potentials and concavity, Math. Ann., 272 (1985), 155–

160.

[4] S.S. DRAGOMIRANDG. KEADY, A Hadamard-Jensen inequality and an ap- plication to the elastic torsion problem, Applicable Analysis, 75 (2000), 285–

295.

[5] R.J. GARDNER, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc., 39 (2002), 355–405.

[6] H. HADWIGER, Konkave eikerperfunktionale und hoher tragheitsmomente, Comment Math. Helv., 30 (1956), 285–296.

[7] G. KEADYANDA. McNABB, The elastic torsion problem: solutions in con- vex domains, N.Z. Journal of Mathematics, 22 (1993), 43–64.

[8] H. KNOTHE, Contributions to the theory of convex bodies, Michigan Math. J., 4 (1957), 39–52.

[9] G. PÓLYA AND G. SZEGÖ, Isoperimetric Inequalities of Mathematical Physics, Princeton Univ. Press, 1970.

[10] R.G. SALAHUDINOV, Isoperimetric inequality for torsional rigidity in the complex plane, J. of Inequal. & Appl., 6 (2001), 253–260.