J. Math. Anal. Al. 44 (3) 3 38 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and Alications journal homeage: www.elsevier.com/locate/jmaa Maximal surface area of a convex set in R n with resect to exonential rotation invariant measures Galyna Livshyts Deartment of Mathematics, Kent State University, Kent, OH 444, USA a r t i c l e i n f o a b s t r a c t Article history: Received 3 August Available online 5 March 3 Submitted by Jesus Bastero Keywords: Convex bodies Convex olytoes Surface area Gaussian measures Let be a ositive number. Consider the robability measure γ with density ϕ (y) = c n, e y. We show that the maximal surface area of a convex body in R n with resect to γ is asymtotically equivalent to C()n 3 4, where the constant C() deends on only. This is a generalization of results due to Ball (993) [] and Nazarov (3) [9] in the case of the standard Gaussian measure γ. 3 Elsevier Inc. All rights reserved.. Introduction As usual, denotes the norm in Euclidean n-sace R n, and A stands for the Lebesgue measure of a measurable set A R n. We will write B n = {x Rn : x } for the unit ball in R n and S n = {x R n : x = } for the unit n-dimensional shere. We will set ν n = B n = π n /Γ n +. A convex body is a comact convex set with nonemty interior. In this aer we will study the geometric roerties of measures γ on R n with density ϕ (y) = c n, e y, where (, ) and c n, is the normalizing constant, chosen in such a way that the measure γ is a robability measure. Many interesting results are known for the case =, where γ becomes the standard Gaussian measure. One should mention the Gaussian isoerimetric inequality of Borell [3] and Sudakov and Tsirel son []: fix some a (, ) and ε > ; then among all measurable sets A R n with γ (A) = a, the set for which γ (A + εb n ) has the smallest Gaussian measure is the half-sace. We refer the reader to the books [,7] for more roerties of the Gaussian measure and inequalities of this tye. Mushtari and Kwaien asked a question in the reverse direction to the isoerimetric inequality: How large can the Gaussian surface area of a convex set A R n be? In [] it was shown that the Gaussian surface area of a convex body in R n is asymtotically bounded from above by Cn 4 as n tends to infinity, where C is an absolute constant. Nazarov [9] gave the comlete solution to this roblem by roving the sharness of Ball s result:.8n 4 su γ ( ).64n 4, where the suremum is taken over all n-dimensional convex sets. Further estimates for γ ( ) were very recently roved by Kane in [6]. Suorted in art by US National Science Foundation grant DMS-65684. E-mail address: glivshyt@kent.edu. -47X/$ see front matter 3 Elsevier Inc. All rights reserved. htt://dx.doi.org/.6/j.jmaa.3.3.4
3 G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 We should note that isoerimetric inequalities for rotation invariant measures were studied by Sudakov and Tsirel son []. For a ositive convex function h(t), let γ be the measure with density e h(log x ). For a >, set M Q (a) = γ (aq ). In [] it was roved that M () Q exists whenever Q is a convex body, and the minimum of M Q () over all convex bodies is attained when Q is a half-sace. Of course, their result can be alied to the measure γ by setting h(t) = et. Some interesting results for manifolds with density were also rovided by Bray and Morgan [4] and further generalized by Maurmann and Morgan [8]. The main goal of this aer is to comlement the study of the isoerimetric roblem for rotation invariant measures and rove an inverse isoerimetric inequality for γ. This is done using a generalization of Nazarov s method from [9]. We recall that the surface area of a convex body Q with resect to the measure γ is defined to be γ ( ) = lim inf ϵ + γ ((Q + ϵb n) \ Q ). ϵ One can also rovide an integral formula for γ ( ): γ ( ) = ϕ (y)dσ (y) = c n, e y dσ (y), () where dσ (y) stands for Lebesgue surface measure. We refer the reader to [6] for the roof for the case =, and the roof given there readily generalizes to other. The following theorem is the main result of this aer: Theorem. For any ositive, c()n 3 4 su γ ( ) C()n 3 4, where C() and c() are ositive constants deending on only. The suremum is taken over all n-dimensional convex sets. In this aer we will denote an asymtotic equality by and an asymtotic inequality by. Namely, A(n) B(n) iff A B( + o()), where o() is an infinitely small number, while n (or some arameter in the context) tends to infinity. Similarly, A(n) B(n) iff A(n) B(n) A(n). Throughout the aer, c, c, C, C, C,... denote absolute constants, indeendent of n and, whose value may change from line to line; C(), c() denote constants deending on the argument only. Using the trick from [,. 43, Proosition ] one can easily derive an estimate γ ( ) e n for any convex set. The calculation is given in Section, as well as some other imortant reliminary facts. The uer bound from Theorem is obtained in Section 3, and the lower bound is shown in Section 4.. Preliminary lemmas We recall that γ is a robability measure on R n with density ϕ (y) = c n, e y, where (, ). The normalizing constant c n, equals [nν n J n, ], where J a, = t a e t dt. (3) We will use an asymtotic estimate for J a, (see Lemma 3 below). This estimate follows immediately from the well-known asymtotic formulas for the Gamma function, which can be obtained by the Lalace method (see, for examle, [5]). In this aer we will rovide several calculations in the sirit of this technique. For the sake of comleteness, we shall resent a short overview of the method here: Lemma. Let h(x) C ([a, b]), where a and b may be infinities, and (a, b). Let be the global maximum oint of h(x) and assume for convenience that h() =. Assume that for any δ > there exists η(δ) > s.t. h(x) < η(δ) for all x [ δ, δ]. Assume also that h () < b and that the integral a eh(x) dx <. Then b e th(x) dx π h ()t, t. a Proof. First, by the conditions of the lemma and the Taylor formula, for a sufficiently small ϵ > there exists a ositive δ = δ(ϵ) such that for any x ( δ, δ) it holds that h(x) h ()x ϵx. ()
G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 33 Thus the integral δ e th(x) dx δ δ (h ()+ϵ) (h () + ϵ) e ty dy δ (h ()+ϵ) π (h () + ϵ)t. (4) Note that for any constant C >, e ty dy e (t )C e y dy = C e C t, (5) C and thus, as t, C δ e th(x) dx δ δ (h () ϵ) (h () ϵ) e ty dy δ (h () ϵ) π (h () ϵ)t. (6) b It remains to show that a eth(x) dx can be asymtotically estimated by the integral over the small interval about zero. Indeed, for an arbitrary ϵ we have chosen δ(ϵ), and now by the condition of the lemma, we can ick η(δ) = η(ϵ), so b e th(x) e (t )η(δ) e h(x) dx = C e C t. (7) (a, δ) (δ,b) Thus, by (4) and (6), π (h () ϵ)t b a a e th(x) π dx (h () + ϵ)t. Taking ϵ small enough we finish the roof. We will now aly Lalace s method to deduce the asymtotic estimate for J a,. Lemma. Let >. Then π J a, a a a e a, as a. Proof. We notice that t a e t dt = a a e a e a log t a t a + dt = a a e a a e a h(x) dx, where h(x) = log x x +. It is easy to check that all the conditions of Lemma are satisfied: h() = h () = ; h () < and e h(x) dx <. In addition, for any δ >, there exists η(δ) such that h(x) < η(δ) for all x [ δ, + δ]. We aly Lemma to finish the roof. Next, we would like to show that the integral t a e t dt (+C)a is an exonentially small function as a for any absolute ositive constant C. Indeed, (+C)a t a e t dt = a a e a a e a h(x) dx, +C
34 G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 where h(x) = log x x +. We note that for any x > + C, h(x) < h( + C) C for some ositive constant C. Thus, the above integral can be estimated with a a e a a e C a +C e h(x) dx. Alying Lemma together with (8) and the fact that e h(x) dx converges to some constant, we obtain the following fact: t a e t dt C ()J a, e C()a. (9) (+C)a Next, we shall observe that the surface area is mostly concentrated in a narrow annulus. Define = (e). Let A = ( + )(n ) B n \ ( )(n ) B n ; we shall call A the concentration annulus. Lemma 3. There exist ositive constants C () and C (), such that γ ( A c ) C ()e C ()n for any convex body Q R n. Proof. Let Q = Q ( )(n ) B n. Then, γ ( y ) = dσ (y) nν n J n, nν n J n, e ( ) n (n ) n nν n nν n J n, = ( ) n (n ) which is exonentially small by the choice of. J n, Define now the surface M = \ ( + )(n ) B n. We can estimate γ (M) using a trick from []. Notice that e y = Consider y M and t Thus t e t dt = t e t χ[ t,t] ( y )dt, y R n. y, ( + )(n ) e y = t e t (+ )(n ) χ[ t,t] ( y )dt. γ (M) = nν n J n, = nν n J n, = nν n J n, J n, C ()e C ()n, e y dσ (y) M M (+ )(n ) (+ )(n ) t n+ e t dt (+ )(n ) n ; then χ [ t,t] ( y ) = and t e t χ[ t,t] ( y )dtdσ (y) t e t M tb n dt where the last inequality follows from (9). This finishes the roof. Remark. Lemma 3 imlies that when obtaining any olynomial bounds on the γ -surface area of the convex body Q R n, it is enough to consider only the ortion of which is contained in the concentration annulus. We can also obtain a rough bound for the γ -surface area of a convex body. Namely, γ ( ) = e x dx = t e t tb n nν n J n, nν n J dt n, J n+, J n, n, n. This bound is far from the best ossible. The next section is devoted to the best ossible asymtotic uer bound. (8)
G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 35 3. The uer bound We will use the aroach develoed by Nazarov in [9]. Let us consider a olar coordinate system x = X(y, t) in R n with y, t >. Then ϕ (y)dσ (y) = D(y, t)ϕ (X(y, t))dσ (y)dt, R n where D(y, t) is the Jacobian of x X(y, t). Define Then and thus ξ(y) = ϕ = (y) D(y, t)ϕ (X(y, t))dt. ϕ (y)ξ(y)dy, ϕ (y)dy min ξ(y). y Following [9], we shall consider two such systems. We will be using subindices for X, D and ξ to distinguish between the two corresonding systems. 3.. The first coordinate system Consider the radial olar coordinate system X (y, t) = yt. The Jacobian is D (y, t) = t n y α, where α = α(y) denotes the absolute value of the cosine of the angle between y and ν y (the unit outer normal vector at y). From (), ξ (y) = e y α y n J n, π e y α y n n e F (n ), as n, () where F(t) = (n ) log t t. Since (n ) is the maximum oint for F(t), we get that F y R n. So we can estimate () from below by π ξ (y) n α. () () (n ) F( y ) for all (3) 3.. The second coordinate system Now consider the normal olar coordinate system X (y, t) = y + tν y. Then D (y, t) for all y Q. Thus, by the law of cosines, Thus, y + tν y y + t + t y α. ξ (y) e y e ( y +t +t y α) dt. Note that for any ositive function f (x) defined on the interval I, and any t I, I Consider e f (t) dt e f (t ) {t : f (t) < f (t )} I. (5) f (t) = ( y + t + t y α). (4)
36 G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 By the intermediate value theorem there is t such that ( y + t + t y α) = y +. Since f (t) is increasing, from (5) and (6) we get ξ (y) e t. Now we need to estimate t from below. By Remark, it suffices to consider y A. For such y, note that the Mean Value Theorem yields y + ( y + ) y. Using (6) and (7), we get α y y + ( y + ) α y t = α y + y α y. Multilying the last exression by its conjugate and alying the inequality a + b a + b, we get ξ (y) e y +. α y Now we shall combine estimates (3) and (8) with the assumtion that y A : ξ(y) := ξ (y) + ξ (y) n π α + C () α. (9) n + Note that (9) is minimized whenever α = C ()n 4. The minimal value of (9) is C() n 3 4. Together with (), this imlies that γ ( A ) C()n 3 4. One can note that C() tends to infinity while tends to infinity or to zero. Finally, we use the above together with Remark to finish the roof of the uer bound art of Theorem. 4. The lower bound Let us consider N uniformly distributed indeendent random vectors x i S n. Let ρ = n 4 and r = r w = n + w, where w [ W, W], and W = n. Consider a random olytoe Q in R n, defined as follows: Q = {x R n : x, x i ρ, i =,..., N}. Let q(t) be the robability that the fixed oint on the shere of radius t + ρ is searated from the origin by the hyerlane x, x i = ρ. Consider a oint x belonging to the intersection of t + ρ S n and the hyerlane x, x = ρ. Since the vectors x,..., x N are indeendent, the robability that x is not searated from the olytoe by any of the remaining N hyerlanes is ( q(t)) N. So the exected value of γ ( ) is N ex ( y + ρ ) ( q( y )) N dy, () nν n J n, R n Passing to olar coordinates, making a change of variables and truncating the integral, we shall estimate () asymtotically from below by ν W n N n n + w e ν n J n, W Note that ν n since w W = n ν n n C n e n N π. Also note that n +w +ρ (6) (7) (8) ( q(r w )) N dw. () n + w n C n n e wn for a ositive constant C deending on only. Using the above facts together with Lemma, we estimate () from below by n +w +ρ W e W e wn ( q(r w )) N dw. ()
G. Livshyts / J. Math. Anal. Al. 44 (3) 3 38 37 Claim. n +w +ρ e C()e n e n e wn. Proof. Plugging in the value n 4 for arameter ρ, and estimating w with W = n, as announced earlier, we obtain that n +w +ρ e e n e n +wn +n +n. The claim now follows from the fact (obtained by the Mean Value Theorem) that + wn + n + n n + wn + O. n Using the above claim we get that () is greater than C ()n e n N W ( q(r w )) N dw. (3) W Next, we estimate the robability q(r) the same way as in [9]: the robability that a oint on a shere is searated from the olytoe by a hyerlane is the quotient of the area of the ca to the area of the whole shere. We recover both of these by integrating over the circles (the sheres with one fewer dimension) and using the Fubini Theorem. Thus, r +ρ q(r) = t r + ρ r +ρ n 3 dt By the Lalace method, the first integral is aroximately equal to πn. r +ρ n 3 t dt. (4) ρ r + ρ Using the elementary inequality a e a e a for all a > (which is true since the function f (a) = e a a is convex for all ositive a, its derivative at zero is, and its value at zero is ), one can estimate the second integral in (4) by ex n 3 4(r + ρ ) t4 ex n 3 t dt ex n 3 r + ρ 4(r + ρ ) ρ4 ex n 3 t dt. r + ρ ρ The first multile is of order e 4 under our assumtions on r and ρ. The second integral can be estimated with usage of the inequality e a t dt ρ ρ aρ e a, a >, ρ >. We note that under our assumtions on ρ and W, aρ = n 3 ρ is of order n ρ +r 3n u to an additive error n. Hence, one can estimate q(r): ρ q(r) Cn 4 e n. (5) Now, one can choose N = Cn 4 e n. From (3) and (5) it follows that the exectation of γ ( ) is greater than C ()n e n n n 4 e W. (6) Plugging in W = n, we get c()n 3 4 for the lower bound for the exectation of γ ( ), which finishes the roof of Theorem. Acknowledgments I would like to thank Artem Zvavitch and Fedor Nazarov for introducing me to the subject, suggesting this roblem to me and extremely helful and fruitful discussions. I would also like to thank the anonymous reviewer for useful suggestions.
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