A NOTE ON THE M^{∗}-LIMITING CONVOLUTION BODY

antonis tsolomitis

Abstract. We introduce the mixed convolution bodies of two convex symmetric bodies. We prove that if the boundary of a body K is smooth enough then as δ tends to 1 the δ–M^{∗}–convolution body of K with itself tends to a multiple of the Euclidean ball after proper normalization. On the other hand we show that the δ–M^{∗}–convolution body of the n–dimensional cube is homothetic to the unit ball of ℓ_{1}^{n}.

1. Introduction

Throughout this note K and L denote convex symmetric bodies in ℝ^{n}. Our notation will be the standard notation that can be found, for example, in [2] and [4]. For 1 ≤ m ≤ n, V_{m}(K) denotes the m–th mixed volume of K (i.e. mixing m copies of K with n − m copies of the Euclidean ball B_{n} of radius one in ℝ^{n}). Thus if m = n then V_{n}(K) = vol_{n}(K) and if m = 1 then V_{1}(K) = w(K) the mean width of K.

For 0 < δ < 1 we define the m–th mixed δ–convolution body of the convex symmetric bodies K and L in ℝ^{n}:

Definition 1.2. The m–th mixed δ–convolution body of K and L is defined to be the set,

C_{m}(δ; K, L) = {x ∈ ℝ^{n} : V_{m} (K ∩ (x + L)) ≥ δV_{m}(K ∩ L)}.

It is a consequence of Brunn–Minkowski inequality for mixed volumes that these bodies are convex.

If we write h(u) for the support function of K in the direction u ∈ 𝕊^{n−1} then we have,

(1.1)  w(K) = 2M_{K}^{∗} = 2 ∫_{𝕊^{n−1}} h(u)dν(u),

where ν is the Lebesgue measure of ℝ^{n} restricted on 𝕊^{n−1} and normalized so that ν(𝕊^{n−1}) = 1. In this note we study the limiting behavior of C_{1}(δ; K, K) (which we will abbreviate with C_{1}(δ)) as δ tends to 1 and K has a C_{+}^{2} boundary. For simplicity we will call C_{1}(δ) the “δ–M^{∗}–convolution body of K”.

Typeset by AMS-TEX

Research at MSRI partially supported by NSF grant DMS-9022140.

We are looking for suitable α ∈ ℝ so that the limit

lim_{δ→1^{−}} \frac{C_{1}(δ)}{(1 − δ)^{α}}

exists (convergence in the Hausdorff distance). In this case we call the limiting body “the limiting M^{∗}–convolution body of K”.

We prove that for a convex symmetric body K in ℝ^{n} with C_{+}^{2} boundary the limiting M^{∗}–convolution body of K is homothetic to the Euclidean ball. We also get a sharp estimate (sharp with respect to the dimension n) of the rate of the convergence of the δ–M^{∗}–convolution body of K to its limit. By C_{+}^{2} we mean that the boundary of K is C^{2} and that the principal curvatures of bd(K) at every point are all positive.

We also show that some smoothness condition on the boundary of K is necessary for this result to be true, by proving that the limiting M^{∗}–convolution body of the n–dimensional cube is homothetic to the unit ball of ℓ_{1}^{n}.

We want to thank Professor V.D.Milman for his encouragement and his guidance to this research and for suggesting the study of mixed convolution bodies.

2. The case “bd(K) is a C_{+}^{2} manifold”

In this section we prove the following:

Theorem 2.1. Let K be a convex symmetric body in ℝ^{n} so that bd(K) is a C_{+}^{2} manifold. Then for all x ∈ 𝕊^{n−1} we have,

(2.1.1)  |∥x∥_{\frac{C_{1}(δ)}{1−δ}} − \frac{c_{n}}{M_{K}^{∗}} | ≤ C \frac{c_{n}}{M_{K}^{∗}} (M_{K}^{∗}n(1 − δ))^{2} ,

where c_{n} = ∫_{𝕊^{n−1}} |⟨x, u⟩|dν(u) ∼ 1/\sqrt{n} and C is a constant independent of the dimension n. In particular,

lim_{δ→1^{−}} \frac{C_{1}(δ)}{1 − δ} = \frac{M_{K}^{∗}}{c_{n}} B_{n}.

Moreover the estimate (2.1.1) is sharp with respect to the dimension n

By “sharp” with respect to the dimension n we mean that there are examples (for instance the n–dimensional Euclidean ball) for which the inequality (2.1.1) holds true if “≤” is substituted with “≥” and the constant C is adjusted by a (universal) constant factor.

Before we proceed with the proof we will need to collect some standard notation which can be found in [4]. We write p : bd(K) → 𝕊^{n−1} for the Gauss map p(x) = N(x) where N(x) denotes the unit normal vector of bd(K) at x. W_{x} denotes the Weingarten map, that is, the differential of p at the point x ∈ bd(K). W_{u}^{−1} is the reverse Weingarten map at u ∈ 𝕊^{n−1} and the eigenvalues of W_{x} and W_{u}^{−1} are respectively the principal curvatures and principal radii of curvature of the manifold bd(K) at x ∈ bd(K) and u ∈ 𝕊^{n−1}. We write ∥W ∥ and ∥W^{−1}∥ for the quantities: sup_{x∈bd(K)} ∥W_{x}∥ and sup_{u∈𝕊^{n−1}} ∥W_{u}^{−1}∥ respectively. These quantities are finite since the manifold bd(K) is assumed to be C_{+}^{2}.

For λ ∈ ℝ and x ∈ 𝕊^{n−1} we write K_{λ} for the set K ∩ (λx + K). p_{λ}^{−1} : 𝕊^{n−1} → bd(K_{λ}) is the reverse Gauss map, that is, the affine hyperplane p_{λ}^{−1}(u) + [u]^{⊥} is tangent to K_{λ} at p_{λ}^{−1}(u). The normal cone of K_{λ} at x is denoted by N(K_{λ}, x) and similarly for K. The normal cone is a convex set (see [4]). Finally h_{λ} will denote the support function of K_{λ}.

Proof. Without loss of generality we may assume that both the bd(K) and 𝕊^{n−1} are equipped with an atlas whose charts are functions which are Lipschitz, their inverses are Lipschitz and they all have the same Lipschitz constant c > 0.

Let x ∈ 𝕊^{n−1} and λ = \frac{1}{∥x∥_{C_{1}(δ)}} ; hence λx ∈ bd (C_{1}(δ)) and

(2.1.2)  M_{K_{λ}}^{∗} = δM_{K}^{∗}.

We estimate now M_{K_{λ}}^{∗}. Let u ∈ 𝕊^{n−1}. We need to compare h_{λ}(u) and h(u). Set Y_{λ} = bd(K) ∩ bd(λx + K).

Case 1. p_{λ}^{−1}(u) ∉ Y_{λ}.

In this case it is easy to see that

h_{λ}(u) = h(u) − |⟨λx, u⟩|.

Case 2. p_{λ}^{−1}(u) ∈ Y_{λ}.

Let y_{λ} = p_{λ}^{−1}(u) and y′_{λ} = y_{λ} − λx ∈ bd(K). The set N(K_{λ}, y_{λ}) ∩ 𝕊^{n−1} defines a curve γ which we assume to be parametrized on [0, 1] with γ(0) = N(K, y_{λ}) and γ(1) = N(K, y′_{λ}). We use the inverse of the Gauss map p to map the curve γ to a curve γ̃ on bd(K) by setting γ̃ = p^{−1}γ. The end points of γ̃ are y_{λ} (label it with A) and y′_{λ} (label it with B). Since u ∈ γ we conclude that the point p^{−1}(u) belongs to the curve γ̃ (label this point by Γ). Thus we get:

0 ≤ h(u) − h_{λ}(u) = |⟨A⃗Γ, u⟩|.

It is not difficult to see that the cosine of the angle of the vectors A⃗Γ and u is less than the largest principal curvature of bd(K) at Γ times |A⃗Γ|, the length of the vector A⃗Γ. Consequently we can write,

0 ≤ h(u) − h_{λ}(u) ≤ ∥W ∥|A⃗Γ|^{2}.

In addition we have,

|A⃗Γ| ≤ length (γ̃|_{A}^{Γ}) ≤ length (γ̃|_{A}^{B})

= ∫_{0}^{1} |d_{t}γ̃|dt = ∫_{0}^{1} |d_{t}p^{−1}γ|dt

≤ ∥W^{−1}∥length(γ) ≤ \frac{2}{π} ∥W^{−1}∥|p(y_{λ}) − p(y′_{λ})|,

where | ▪ | is the standard Euclidean norm. Without loss of generality we can assume that the points y_{λ} and y′_{λ} belong to the same chart at y_{λ}. Let φ be the chart mapping ℝ^{n−1} to a neighborhood of y_{λ} on bd(K) and ψ the chart mapping ℝ^{n−1} on a neighborhood of N(K, y_{λ}) in 𝕊^{n−1}. We assume, as we may, that the graph of γ is contained in the range of the chart ψ. It is now clear from the above series of inequalities that

|A⃗Γ| ≤ c_{0}∥W^{−1}∥|ψ^{−1}pφ(t) − ψ^{−1}pφ(s)|,

where t and s are points in ℝ^{n−1} such that φ(t) = y_{λ} and φ(s) = y′_{λ} and c_{0} > 0 is a universal constant. Now the mean value theorem for curves gives,

|A⃗Γ| ≤ C∥W^{−1}∥∥W ∥|t − s|
≤ C∥W^{−1}∥∥W ∥|y_{λ} − y′_{λ}|
= C∥W^{−1}∥∥W ∥λ,

where C may denote a different constant every time it appears. Thus we have,

0 ≤ h(u) − h_{λ}(u) ≤ C∥W ∥ (∥W^{−1}∥∥W ∥)^{2} λ^{2}.

Consequently,

∫_{𝕊^{n−1}\p_{λ}(Y_{λ})} (h(u) − |⟨λx, u⟩|) dν(u) + ∫_{p_{λ}(Y_{λ})} (h(u) − Cλ^{2}) dν(u)
≤ M_{K_{λ}}^{∗} = δM_{K}^{∗} ≤
∫_{𝕊^{n−1}\p_{λ}(Y_{λ})} (h(u) − |⟨λx, u⟩|) dν(u) + ∫_{p_{λ}(Y_{λ})} h(u)dν(u),

where C now depends on ∥W ∥ and ∥W^{−1}∥.

Rearranging and using c_{n} for the quantity ∫_{𝕊^{n−1}} |⟨x, u⟩|dν(u) and the fact λ = 1/∥x∥_{C_{1}(δ)} we get:

|∥x∥_{\frac{C_{1}(δ)}{1−δ}} − \frac{c_{n}}{M_{K}^{∗}} | ≤ \frac{c_{n}}{M_{K}^{∗}} (\frac{∫_{p_{λ}(Y_{λ})} |⟨x, u⟩|dν(u)}{c_{n}} + Cλ\frac{μ (p_{λ}(Y_{λ}))}{c_{n}}) .

We observe now that for u ∈ p_{λ}(Y_{λ}), |⟨x, u⟩| ≤ length(γ)/2 ≤ ∥W ∥λ. Using this in the last inequality and the fact that p_{λ}(Y_{λ}) is a “band” around an equator of 𝕊^{n−1} of width at most length(γ)/2 we get,

|∥x∥_{\frac{C_{1}(δ)}{1−δ}} − \frac{c_{n}}{M_{K}^{∗}} | ≤ \frac{c_{n}}{M_{K}^{∗}} Cnλ^{2}

≤ \frac{c_{n}}{M_{K}^{∗}} Cn\frac{(1 − δ)^{2}}{∥x∥2_{\frac{C_{1}(δ)}{1−δ}}} .

Our final task is to get rid of the norm that appears on the right side of the latter inequality. Set

T = \frac{∥x∥_{C_{1}(δ)/1−δ}}{c_{n}/M_{K}^{∗}} .

We have shown that

T^{2}|T − 1| ≤ C \frac{M_{K}^{∗}}{c_{n}} n(1 − δ)^{2}.

If T ≥ 1 then we can just drop the factor T^{2} and we are done. If T < 1 we write T^{2}|T − 1| as (1 − (1 − T))^{2} (1 − T) and we consider the function

f(x) = (1 − x)^{2}x : (−∞, \frac{1}{3}) → ℝ.

This function is strictly increasing thus invertible on its range, that is, f^{−1} is well defined and increasing in (−∞, \frac{4}{27}). Consequently if

(2.1.3)  C \frac{M_{K}^{∗}}{c_{n}} n(1 − δ)^{2} ≤ \frac{4}{27},

we conclude that,

0 ≤ 1 − T ≤ f^{−1} (C \frac{M_{K}^{∗}}{c_{n}} n(1 − δ)^{2})

≤ C \frac{M_{K}^{∗}}{c_{n}} n(1 − δ)^{2}.

The last inequality is true since the derivative of f^{−1} at zero is 1. Observe also that the convergence is “essentially realized” after (2.1.3) is satisfied. □

We now proceed to show that some smoothness conditions on the boundary of K are necessary, by proving that the limiting M^{∗}–convolution body of the n– dimensional cube is homothetic to the unit ball of ℓ_{1}^{n}. In fact we show that the δ–M^{∗}–convolution body of the cube is already homothetic to the unit ball of ℓ_{1}^{n}.

Example 2.3. Let P = [−1, 1]^{n}. Then for 0 < δ < 1 we have,

C_{1}(P) = \frac{C_{1}(δ; P, P)}{1 − δ} = n^{3/2}vol_{n−1}(𝕊^{n−1})B_{ℓ_{1}^{n}}.

Proof. Let x = ∑_{j=1}^{n} x_{j}e_{j} where x_{j} ≥ 0 for all j = 1, 2, …, n and e_{j} is the standard basis of ℝ^{n}. Let λ > 0 be such that λx ∈ bd (C_{1}(δ)). Then,

P ∩ (λx + P) = {y ∈ ℝ^{n} : y = ∑_{j=1}^{n} y_{i}e_{i}, −1 + λx_{j} ≤ y_{j} ≤ 1, j = 1, 2, …, n}.

The vertices of P_{λ} = P ∩ (λx + P) are the points ∑_{j=1}^{n} α_{j}e_{j} where α_{j} is either 1 or −1 + λx for all j. Without loss of generality we can assume that −1 + λx_{j} < 0 for all the indices j. Put sign α_{j} = α_{j}/|α_{j}| when α_{j} ≠ 0 and sign 0 = 0. Fix a sequence of α_{j}’s so that the point v = ∑_{j=1}^{n} α_{j}e_{j} is a vertex of P_{λ}. Clearly,

N (P_{λ}, v) = N (P, ∑_{j=1}^{n}(sign α_{j})e_{j}) .

If u ∈ 𝕊^{n−1} ∩ N(P_{λ}, v) then,

h_{λ}(u) = h(u) − |⟨∑_{j=1}^{n}(α_{j} − sign α_{j})e_{j}, u⟩|.

If sign α_{j} = 1 then α_{j} − sign α_{j} = 0 otherwise α_{j} − sign α_{j} = λx.

Let A ⊆ {1, 2, …, n}. Consider the “A–orthant”:

O_{A} = {y ∈ ℝ^{n} : ⟨y, e_{j}⟩ < 0, if j ∈ A and ⟨y, e_{j}⟩ ≥ 0 if j ∉ A}.

Then O_{A} = N (P, ∑_{j=1}^{n}(sign α_{j})e_{j}) if and only if sign α_{j} = 1 exactly for every j ∉ A. Hence,

h_{λ}(u) = h(u) − |⟨∑_{j∈A} λx_{j}e_{j}, u⟩|,

for all u ∈ O_{A} ∩ 𝕊^{n−1}. Hence using the facts M_{P_{λ}}^{∗} = δM_{P}^{∗} and λ = 1/∥x∥_{C_{1}(δ)} we get,

∥x∥_{\frac{C_{1}(δ)}{1−δ}} = −\frac{1}{M_{P}^{∗}} ∑_{A⊆\{1,2,...,n\}} ∑_{j∈A} x_{j} ∫_{O_{A}∩𝕊^{n−1}}⟨e_{j}, u⟩dν(u),

which gives the result since

∫_{O_{A}∩𝕊^{n−1}}⟨e_{j}, u⟩dν(u) = \frac{1}{2^{n−1}} ∫_{𝕊^{n−1}} |⟨e_{1}, u⟩|dν(u).

□

References

1. Kiener, K., Extremalität von Ellipsoiden und die Faltungsungleichung von Sobolev, Arch. Math. 46 (1986), 162–168.

2. Milman, V. and Schechtmann, G., Asymptotic theory of finite dimensional normed spaces, vol. 1200, Springer Lecture Notes, 1986.

3. Schmuckenschläger, M., The distribution function of the convolution square of a convex symmetric body in ℝ^{n}, Israel Journal of Mathematics 78 (1992), 309–334.

4. Schneider, R., Convex bodies: The Brunn–Minkowski theory, vol. 44, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1993.

5. Tsolomitis, A., Convolution bodies and their limiting behavior, Duke J. Math. (submitted).

Department of Mathematics, 231 W.18th Avenue, Columbus, OH 43210

Current address: Mathematical Sciences Research Institute, 1000 Centennial Road, Berkeley, CA 94720

E-mail address: atsol@math.mps.ohio-state.edu