Reverse Triangle Inequalities for Riesz Potentials and Connections ...

Reverse Triangle Inequalities for RieszPotentials and Connections with PolarizationI. E. Pritsker, E. B. Saff and W. WiseAbstractWe study reverse triangle inequalities for Riesz potentials and their connectionwith polarization. This work generalizes inequalities for sup norms of products ofpolynomials, and reverse triangle inequalities for logarithmic potentials. The main toolused in the proofs is the representation for a power of the farthest distance function asa Riesz potential of a unit Borel measure.MSC 2010. Primary 31A15, 52A40; Secondary 31A05, 30C15Key words. Riesz potential, farthest distance function, Riesz decomposition, polarizationinequality, Chebyshev constant1 Introduction: Products of Polynomials and Sums of LogarithmicPotentialsLet E be a compact set in C. For any set of real-valued functions f j , j = 1, . . . , m, we havem∑j=1sup f j ≥ supE Eby the triangle inequality. It is not possible to reverse this inequality for arbitrary functions,even by introducing additive constants. However, by restricting the class of functions we canreverse the inequality with sharp additive constants to obtain expressions of the formj=1m∑j=1m∑j=1sup f j ≤ C + supEEf jm∑f j . (1)We begin by considering logarithmic potentials p ν (z) = ∫ log |z − t| dν(t). Let ν j , j =1, . . . , m, be positive compactly supported Borel measures, normalized so that ν := ∑ mj=1 ν jis a unit measure. We want to find a sharp additive constant C such thatm∑sup p ν jm∑≤ C + sup p ν j= C + sup p ν . (2)EEE1j=1j=1

The motivation for such inequalities comes from inequalities for the norms of productsof polynomials. Let P (z) = ∏ nj=1 (z − a j) be a monic polynomial. Then log |P (z)| =n ∫ ∑log |z − t|dτ(t), where τ = 1 nn j=1 δ a jis the normalized counting measure of the zeros ofP , with δ aj being the unit point mass at a j . Let ||P || E be the uniform (sup) norm on E.Then for polynomials P j , j = 1, . . . , m, inequality (2) can be rewritten as∣∣ m∏∣∣∣∣ ∣∣∣∣ ∏ m||P j || E ≤ M nj=1j=1P j∣ ∣∣∣∣∣ ∣∣∣∣E(3)where M = e C and n is the degree of ∏ mj=1 P j.Kneser [13] found the first sharp constant M for inequality (3). Let E = [−1, 1] andconsider only two factors so that m = 2. Then (3) holds with the multiplicative constantM = 2 n−1ndeg P∏ 1k=1(1 + cos 2k − 12n π ) 1deg P 2n ∏k=1(1 + cos 2k − 1 ) 12n π n. (4)The Chebyshev polynomial shows that this constant is sharp. A weaker result was previouslygiven by Aumann [2]. Borwein [6] provided an alternative proof for this constant (4). Heshowed further that on E = [−1, 1], inequality (3) holds for any number of factors m withmultiplicative constantM = 2 n−1n[ n 2∏]k=1(1 + cos 2k − 1 ) 22n π n. (5)Another series of such constants were found for E = D, the closed unit disk. Mahler[16], building on a weaker result by Gelfond [11, p. 135], showed that (3) holds forM = 2. (6)While the base 2 cannot be decreased, Kroó and Pritsker [14] showed that for m ≤ n, we canuse M = 2 n−1n . Furthermore, Boyd [8, 7] expressed the multiplicative constant as a functionof the number of factors m, and found( ∫m π/m(M = exp log 2 cos t π2) ) .0This constant is asymptotically best possible for each fixed m as n → ∞.For general sets E, the constant M E depends on the geometry of the set. Let E be acompact set of positive capacity. Pritsker [18] showed that a sharp multiplicative constantin (3) is given byM E = exp (∫ log d E (z)dµ E (z) ),cap(E)where µ E is the equilibrium measure of E and d E is the farthest distance function defined byd E (x) := sup t∈E |x − t|, x ∈ C. Note that this constant generalizes several previous results.2

We can calculate that M [−1,1] ≈ 3.20991, which is the asymptotic version of Borwein’s constantfrom (5) as n → ∞. For the closed unit disk, we obtain M D = 2, which is the constantgiven by Mahler (6). Furthermore, Pritsker and Ruscheweyh [20, 21] showed that M D is alower bound on M E for any compact E with positive capacity. They also conjectured thatM [−1,1] is an upper bound for all non-degenerate continua. Shortly afterward, Baernstein,Pritsker, and Laugesen [3] showed M [−1,1] is an upper bound for centrally symmetric continua.The assumption that E has positive capacity is vital. For example, if E is a finiteset, then no inequality of the form (3) is possible for any number of factors m ≥ 2. If E iscountable, then the constant M could grow arbitrarily fast as m grows large.All results for M in (3) apply as well to C in (2) with C = log M, see [22]. Specifically,(2) holds with sharp additive constant∫C E = log d E (z)dµ E (z) − log cap(E).It follows from [20, 21] that C D = log 2 is a lower bound for C E for any compact set E withpositive capacity, while C [−1,1] ≈ log 3.20991 is an upper bound on C E for certain classes ofsets E. Allowing the constant to be dependent on the number of terms m, Pritsker and Saff[22] found that (2) holds for m terms with∫C E (m) = maxc k ∈∂Elog max1≤k≤m |z − c k|dµ E (z) − log cap(E).Note that lim m→∞ C E (m) = C E .These results were generalized to Green potentials by Pritsker [19]. Let p j , j = 1, . . . , m,be Green potentials [1, p. 96] on a domain G ⊂ C. Then for any compact set E ⊂ G wehavem∑j=1infEp j ≥ C + M infEm∑p j (7)where M and C are given in [19] as explicit constants depending only on G and E, and Cis sharp.The outline of the present paper is as follows. In the next section we prove a reversetriangle inequality analogous to (7) for Riesz potentials (see Theorem 2.3), and give severalexamples. The main ingredient in the proofs is the representation of a power of the farthestdistance function as the Riesz potential of a positive unit measure (see Theorem 2.2), whichmay be of independent interest. We consider connections of the reverse triangle inequalitywith polarization inequalities for Riesz potentials in Section 3. Section 4 contains all proofs.2 Riesz Potentials and the Distance FunctionWe now consider a compact set E ⊂ R N , N ≥ 2, and Riesz potentials of the form U µ α(x) =∫|x − t| α−N dµ(t) for 0 < α ≤ 2. For α = 2, these are Newtonian potentials, and they aresuperharmonic in R N , N ≥ 3. If N = α = 2 then one may study inequalities for logarithmic3j=1

potentials as done in [22]. We do not consider the case N = α = 2 in this paper. For 0 < α

everywhere, and we may apply the Riesz Decomposition Theorem for Newtonian potentials.It states that there exists a unique Borel measure σ 2 such that∫d 2−NE(x) = |x − t| 2−N dσ 2 (t) + h(x),. Since d 2−NE(x) is positive and tendsis a Newtonianpotential. The complete proof of Theorem 2.2 is given in Section 4.In the Newtonian case, we can also give explicit examples of σ 2 . If B is the unit ball andd B (x) = |x| + 1, then the measure σ B is found using the Laplacian. Specifically, dσ B (x) =where h is the greatest harmonic minorant of d 2−NEto zero as x → ∞, h must be identically zero [1, p. 106] and hence d 2−NE∆d 2−NB(x) = c|x|N−2 d −NBdS where c is a constant and dS is the surface area measure onthe unit sphere. If L = [−1, 1] then ∆d 2−NL= 0 everywhere except on the hyperplanethat is the perpendicular bisector of the segment. It follows that σ L is supported on thathyperplane [23]. Its value can be calculated using the generalized Laplacian, and is given bydσ L = cd −NLdS where c is a constant and dS is the surface area measure on the hyperplane[23]. For 0 < α < 2, the measure σ E should be calculated using fractional Laplacians.We are now prepared to state a reverse triangle inequality for Riesz potentials.Theorem 2.3. Let E ⊂ R N be a compact set with the minimum α-energy W α (E) < ∞,where 0 < α ≤ 2. Suppose that ν k , k = 1, . . . , m, are positive compactly supported Borelmeasures, normalized so that ν := ∑ mk=1 ν k is a unit measure, with m ≥ 2. Thenm∑k=1infEU ν kα≥ C E (α, m) + infEm∑k=1U ν kα , (9)where∫C E (α, m) := minc k ∈Emin |x − c k| α−N dµ α (x) − W α (E)1≤k≤mcannot be replaced by a larger constant for each m ≥ 2. Furthermore, (9) holds with C E (α, m)replaced by∫C E (α) := d α−NE(x)dµ α (x) − W α (E),which does not depend on m.In the Newtonian case α = 2, the minimum principle holds and so the minimum inC E (2, m) is achieved on the boundary of E. Thus∫C E (2, m) = min min |x − c k| 2−N dµ 2 (x) − W 2 (E).c k ∈∂E 1≤k≤mA closed set S ⊂ E is called dominant ifd E (x) = max |x − t| for all x ∈ supp(µ α).t∈S5

When E has at least one finite dominant set, we define a minimal dominant set D E as adominant set with the smallest number of points denoted by card(D E ). Of course, E mightnot have finite dominant sets at all, in which case we can take any dominant set as theminimal dominant set, e.g., D E = ∂E. For example, let E be a polyhedron. The vertices ofE are a dominant set, since d E (x) = max vertices t ∈ E |x − t| everywhere, not just in supp(µ α ).However, this need not be the minimal dominant set. For example, let E be a pyramid. If theapex is close to the base, then it will not be in the minimal dominant set. The hemispherehas the equator as the smallest dominant set, however this set is infinite.Corollary 2.4. For every m ≥ 2, we have C E (α, m) ≥ C E (α). In particular, if m C E (α), while C E (α, m) = C E (α) for all m ≥ card(D E ). Furthermore,the constants C E (α, m) are decreasing in m and lim m→∞ C E (α, m) = C E (α).Corollary 2.5. If E ⊂ R N is a compact set with C 1 -smooth boundary and with finitely manyconnected components, then C E (α, m) > C E (α) for all m ∈ N, m ≥ 2.If E = L := [−1, 1] ⊂ R 2 and 1 < α < 2, then d L (x) = max(|x − 1|, |x + 1|), x ∈ R 2 ,so that the endpoints form the minimal dominant set with card(D L ) = 2. Thus C L (α) =C L (α, 2) = C L (α, m), m ≥ 2.We finish this section with several explicit examples.Example 2.6 (Unit circle T in C). Let T ⊂ C be the unit circle, and let 1 < α < 2. Weknow that dµ α (e iθ ) = dθ/2π andsee [15]. We prove in Section 4 that∫minc k ∈EW α (T) = 2α−2 Γ( α−1√ π Γ( α) ,22 )min |x − c k| α−2 dµ α (x) = 2 α−2 2m ( π1≤k≤m π I 2mwhere I(x) = ∫ x0 cosα−2 θ dθ. It is obvious that d T (x) = 2, x ∈ T, and that S has no finitedominant set. Therefore,C T (α, m) = 2 α−2 2m ( π)π I − 2α−2 Γ( α−1√ ) 22m π Γ( α) > C T (α) = 2 α−2 − 2α−2 Γ( α−1√ ) 22 π Γ( α) .2Example 2.7 (Unit sphere S N−1 in R N ). Let S N−1 := {x ∈ R N : |x| = 1}, N ≥ 3, and let1 < α ≤ 2. It is known that dµ α = dσ/ω N is the normalized surface area on S N−1 andW α (S N−1 ) = 2α−2 Γ( N 2√ πα−1)Γ( ) 2) ,2Γ( N+α−2see [15]. It is also clear that d S N−1(x) = 2, x ∈ S N−1 , and that S N−1 has no finite dominantset. HenceC S N−1(α, m) > C S N−1(α) = 2 α−N − 2α−2 Γ( N α−1)Γ(2√ ) 2π Γ( N+α−2 ) .26),

Example 2.8 (Unit ball B N in R N ). Let B N := {x ∈ R N : |x| ≤ 1}, N ≥ 2, and let0 < α ≤ 2. Again, B N has no finite dominant set. The Wiener constant of the ball isW α (B N ) =Γ(N−α+22)Γ( α 2 )Γ( N 2 ) ,see [15].If α = 2 and N ≥ 3 then the equilibrium measure of the ball dµ 2 = dσ/ω N is thenormalized surface area on S N−1 = ∂B N , so that d B N (x) = 2, x ∈ S N−1 = supp(µ 2 ). HenceC B N (2, m) > C B N (2) = 2 2−N − 1.If 0 < α < 2 then the equilibrium measure of the ball isdµ α (x) = Γ ( )N−α+22π N/2 Γ ( R)α−N dxfor |x| < R,1 − α (R 2 − |x| 2 )α/22see [15, p. 163]. Since supp(µ α ) = B N in this case, we note that d B N (x) = 1 + |x|, x ∈ B N ,so that∫N−α+2C B N (α, m) > C B N (α) = (1 + |x|) α−N Γ( )Γ( α 2 2dµ α (x) − )Γ( N ) ,2where µ α is given above.3 Connections to Polarization InequalitiesLet E be a compact set in R N and let A m = {x j } m j=1, denote an m-point subset of E. TheRiesz polarization quantities, introduced by Ohtsuka [17] and recently studied by Erdélyiand Saff [9], are given byM s (A m , E) := infx∈Em∑j=1|x − x j | −s and Mm(E) s := sup M s (A m , E), s > 0.A m ⊂ELet ν j denote the normalized point mass δ xj /m, so that ∑ mj=1 ν j is a unit measure. TheRiesz polarization quantity for s = N − α may be rewritten in terms of potentials asM N−α (A m , E) = m infEAs proved by Ohtsuka [17], the normalized limitm∑j=1U ν jα .M s (E) := limm→∞ M s m(E)/mexists as an extended real number and is called the Chebyshev constant of E for the Rieszs-potential. Moreover, he showed that this Chebyshev constant is always greater than orequal to the associated Wiener constant. Combining this fact with Frostman’s theorem wededuce the following:7

Proposition 3.1. For 0 < α ≤ 2 and any compact set E ⊂ R N there holdsM N−α (E) = W α (E). (10)Indeed, given a unit Borel measure µ, Frostman’s theorem for such α and E gives∫ ∫inf U α µ ≤ U α µ dµ α = U µ αα dµ ≤ W α (E),Eso that M N−α (E) ≤ W α (E), which together with Ohtsuka’s inequality yields (10). Alternatively,one can deduce (10) by observing that for the given range of α, a maximum principleholds for the equilibrium potential and appealing to Theorem 11 of Farkas and Nagy [10].Bounds on the quantity Mm N−α (E)/m and the sets A m which achieve the maximumin MmN−α (E) have been the subject of several recent papers [9, 12]. The reverse triangleinequality in Theorem 2.3 is directly connected with MmN−α (E)/m in the case of atomicmeasures. Recall that the inequality (9) holds for arbitrary positive Borel measures ν j suchthat ∑ mj=1 ν j is a unit measure. We now introduce a similar inequality where each ν j = δ xj /mis a point mass 1/m supported at x j ∈ E:1mm∑j=1inf |x − x j| α−N ≥ CE(α, δ m) + 1x∈E m infx∈Em∑|x − x j | α−N ,where CE δ (α, m) denotes the largest (best) constant such that the above inequality holds forall {x j } m j=1 ⊂ E. Clearly, we have CE δ (α, m) ≥ C E(α, m).From the definitions of CE δ N−α(α, m) and Mm(E) we immediately deduce that for allα < N,maxA m⊂E1mm∑j=1d α−NE(x j ) − M mN−α (E)mIn particular, if E is the unit sphere S N−1 ⊂ R N , we have2 α−N − M N−αm (S N−1 )mj=1≥ C δ E(α, m).= C δ S N−1 (α, m). (11)In [12], it is proved that for the unit circle T = S 1 the maximum polarization for any m ≥ 2is attained for m distinct equally spaced points. Moreover, this maximum, which occurs atthe midpoints of the m subarcs joining adjacent points is known explicitly (in finite terms)when N − α is a positive even integer, and asymptotically for all −∞ < α < N. Therebywe obtain the following.Proposition 3.2. For the unit circle T = S 1 there holds, for all −∞ < α < 2,C δ T(α, m) = 2 α−2 − M 2−α (A ∗ m, T)m= 2 α−2 − M m2−α (T)m , (12)where A ∗ m = {e i2πk/m : k = 1, . . . , m}. Moreover the following asymptotic formulas hold asm → ∞ :8

⎧2ζ(2 − α)−(2π) ⎪⎨2−α (22−α − 1)m 1−α , 1 > α > −∞ ,CT(α, δ m) ∼ − 1 π log m , α = 1 ,⎪⎩ 2 α−2 − 2α−2 Γ ( (13))α−12√ π Γ ( ) = Cα T (α) , 1 < α < 2,2where ζ(s) denotes the classical Riemann zeta function and a m ∼ b m means that lim m→∞ a m /b m =1.For 1 < α < 2, we have from Example 2.6 and (13) that, for each m ≥ 1,C T (α) < C T (α, m) ≤ C δ T(α, m),with equality holding throughout in the limit as m → ∞. Consequently, from the formulasin Example 2.6 we have(MmN−α (T)= 2 α−2 −C δmT(α, m) ≤ 2 α−2 −C T (α, m) = W α (T)+2 α−2 1 − 2m ( π) )π I < W α (T).2mWe remark that the inequality MmN−α (T) ≤ mW α (T) was found by a different method in(3.7) of [9].Utilizing (12) and the polarization formulas in [12], we list the first few explicit formulasfor CT δ (α, m) that hold whenever α is a nonpositive even integer and m ≥ 1:C δ T(0, m) = 1 4 − m 4 ,C δ T(−2, m) = 1 16 − m 24 − m348 ,C δ T(−4, m) = 164 − m120 − m3192 − m5480 .For the unit sphere in higher dimensions, we have the following.Proposition 3.3. For the unit sphere S N−1 , N > 2, in R N equation (11) holds for all−∞ < α < N. Moreover, the following asymptotic formulas hold as m → ∞ :⎧( ) (N−α)/(N−1) Γ(N/2)−σ(N − α, N − 1)m 1−α⎪⎨2π N/2N−1 , 1 > α > −∞ ,C δ S(α, m) ∼N−1 − log m Γ(N/2)√ π (N − 1)Γ((N − 1)/2) , α = 1 ,⎪⎩ 2 α−N − W α (S N−1 ) = C S N−1(α) , 1 < α < N,(14)where σ(N − α, N − 1) is a positive constant that depends only on α and N (cf. [5]), andwhere the formulas for W α (S N−1 ) and C S N−1(α) are given in Example 2.7.9

For the unit ball we have the following result.Proposition 3.4. For the unit ball B N in R N there holds, for all −∞ < α < N,1 − M N−αm (B N )mMoreover the following asymptotic formulas hold as m → ∞ :≥ C δ B(α, m) ≥ 2 α−N − M m N−α (B N ). (15)Nm⎧( ) (N−α)/N⎪⎨Γ(1 + N/2)C δ −σ(N − α, N)m −α/N , 0 > α > −∞ ,B(α, m) ∼π N⎪⎩N/2 − log m , α = 0 ,(16)where σ(N − α, N) is a positive constant that depends only on α and N.We remark that asymptotic formulas similar to those in Proposition 3.4 can be obtainedfor C δ E (α, m) for a large class of N-dimensional subsets of RN by appealing to the results in[4] and [5].4 ProofsWe begin with a lemma that will be used in the proof of Theorem 2.3. Let F n = {x k,n } n k=1be a set of n points in E. Let τ n be their normalized counting measure and let 0 < α < N.We define the discrete α-energy of τ n byE α [τ n ] :=2n(n − 1)∑1≤j

Since supp(µ α ) ⊂ E, this impliesinfEU τ nα≤ W α (E).On the other hand, for the (n + 1)-tuple (x, ξ 1,n , . . . , ξ n,n ) ⊂ E we may again apply theextremal property of F n to obtain∑1≤j

for the averaging measure ε (r)α used in the definition of α-superharmonicity in [15, p. 112].Furthermore, Property (i) [15, p. 114] for α-superharmonic functions gives thatUασα(x) = lim Uασα∗ ε (r)α (x) = lim d α−Nr→0 r→0E∗ ε (r)α (x) = d α−NE(x), x ∈ R N ,where we used the fact that ε (r)α∗→ δ 0 as r → 0 [15, p. 112] on the last step.We now consider the mass of the measure σ α . Assume, without loss of generality, thatthe origin is a point in E. Consider the ball B(R) of radius R > diam(E) about the origin.We average∫d α−NE(x) = |x − t| α−N dσ α (t)R Nwith respect to the α-equilibrium measure τ R of the ball B(R), to obtain∫∫ ∫M(R) := d α−NE(x)dτ R (x) = |x − y| α−N dσ α (y)dτ R (x). (17)B(R)B(R) R NWe begin with the case α = 2, for which the α-equilibrium measure is the normalizedsurface area measure dτ R = dx/(ω N R N−1 ). It is a standard fact [1, p. 100] that the potentialof the equilibrium measure is given byU τ R2 (x) =∫1ω N R N−1|x − t| 2−N dt ={R 2−N if |x| ≤ R,|x| 2−N if |x| > R.Consider the left hand side of (17). We know R ≤ d E (x) ≤ R + diam(E) on supp(τ R ) =∂B(R). Thus(R + diam(E)) 2−N ≤ M(R) ≤ R 2−N . (18)On the other hand, we may apply Tonelli’s Theorem to the right hand side of (17) andobtain∫ ∫∫|x − t| 2−N dσ 2 (t)dτ R (x) = |x − t|∂B(R) R∫R dτ R (x)dσ 2 (t)N ∂B(R)∫= U τ R2 (y)dσ 2 (y)∫R N ∫= U τ R2 (t)dσ 2 (t) + U τ R2 (t)dσ 2 (t)|t|R= R 2−N σ 2 (B(R)) + U τ R2 (t)dσ 2 (t).Since 0 < U τ R2 (t) < R 2−N for |t| > R, we have|t|>RR 2−N σ 2 (B(R)) ≤ M(R) ≤ R 2−N σ 2 (R N ).Combining the above inequality with (18) and letting R → ∞, we obtain σ 2 (R N ) = 1.12

The proof in the case of 0 < α < 2 is similar. In this case, the α-equilibrium measure isgiven in [15, p. 163] aswhere A is the constantdτ R (x) = AR α−N (R 2 − |x| 2 ) −α/2 dx for |x| < R,A =Its potential U τ R α (x) = ∫ |x − t| α−N dτ R (t) isfor all |y| ≤ R [15, (A.1)]. Using the fact thatwe calculate for |y| ≤ R thatIntroducing the notationΓ ( N−α+ 1 )2π N/2 Γ ( ).1 − α 2U τ Rα (y) = AR α−N π N/2+1Γ(N/2) sin(πα/2) ,πsin(πx)U τ Γ ((N − α)/2 + 1)Rα (y) =π N/2 Γ (1 − α/2) Rα−N== Γ(x)Γ(1 − x),Γ(α/2)Γ ((N − α)/2 + 1)Γ(N/2)π N/2+1c(N, α) = AΓ(N/2) sin(πα/2)πN/2Γ(α/2)Γ(1 − α/2)Γ(N/2)R α−N .Γ(α/2)Γ ((N − α)/2 + 1)= ,Γ(N/2)we obtainU τ Rα (y) = c(N, α)R α−N , for all |y| ≤ R. (19)Furthermore, this same value serves as the upper bound of the potential for all |y| > R.Notice that c(N, 2) = 1 and hence (19) is a generalization of the fact that U τ R2 (x) = R 2−Nfor |x| ≤ R when α = 2.Consider the left hand side of (17). We know |x| ≤ d E (x) ≤ |x| + diam(E) in B(R). Weuse the lower bound on d E to find an upper bound on M(R). Applying the calculations in[15, Appendix] again, we conclude that∫∫d α−NE(x)dτ R (x) ≤ |x| α−N dτ R (x)B(R)B(R)∫= AR α−N (R 2 − |x| 2 ) −α/2 |x| α−N dxB(R)= AR α−N π N/2+1Γ(N/2) sin(πα/2)= c(N, α)R α−N .13

Next we use the upper bound on d E to obtain a lower bound for M(R). Let d = diam(E).Then for any ϵ > 0 we have d ≤ ϵ|x| for any x not in B(d/ϵ). Hence∫∫d α−NE(x)dτ R (x) ≥ (|x| + d) α−N dτ R (x)B(R)B(R)∫>(|x| + d) α−N dτ R (x)B(R)\B(d/ϵ)∫≥|x| α−N (1 + ϵ) α−N dτ R (x)B(R)\B(d/ϵ)∫= (1 + ϵ) α−N U τ Rα (0) − (1 + ϵ) α−N |x| α−N dτ R (x)B(d/ϵ)∫= (1 + ϵ) α−N c(N, α)R α−N − (1 + ϵ) α−N |x| α−N dτ R (x)Estimating the integral over the ball B(d/ϵ), we find∫B(d/ϵ)B(d/ϵ)∫ d/ϵ|x| α−N dτ R (x) =R α−N Aω N |x| α−1 (R 2 − |x| 2 ) −α/2 d|x|0≤R α−N (R 2 − d2ϵ 2 ) −α/2Aω N d ααϵ α .Since the above integral is also bounded below by zero, it follows that it is O(R −N ) and thus(1 + ϵ) α−N c(N, α)R α−N − O(R −N ) < M(R) ≤ c(N, α)R α−N . (20)On the other hand, we may apply Tonelli’s Theorem on the right hand side of (17) toobtain∫B(R)∫|x − y| α−N dσ α (y)dτ R (x) =R N ∫=∫=∫R N ∫U τ RαR N|y|≤R|x − y| α−N dτ R (x)dσ α (y)B(R)(y)dσ α (y)Applying the calculation of the potential in (19), we findU τ Rα (y)dσ α (y) +∫|y|>RU τ Rα (y)dσ α (y).c(N, α)R α−N σ α (B(R)) ≤ M(R) < c(N, α)R α−N σ α (R N ). (21)Combining (20) and (21), dividing by R α−N and then letting R → ∞, we obtain(1 + ϵ) α−N ≤ σ α (R N ) ≤ 1.Finally, we conclude σ α (R N ) = 1 by letting ϵ → 0.14

Proof of Theorem 2.3. For any positive Borel measure µ, the potential∫U α(t) µ = |t − x| α−N dµ(x)is lower semicontinuous [15, p. 59], and hence attains its infimum on the compact set E.Thus we may choose c k ∈ E such thatfor each k = 1, . . . , m. It follows thatm∑inf U ν kα =Ek=1=infEm∑k=1U ν kαm∑∫k=1= U ν kα (c k )U ν kα (c k )|c k − x| α−N dν k (x)∫≥ min |c k − x| α−N dν(x)1≤k≤m∫ () α−N= max |c k − x| dν(x).1≤k≤mThe function d m (x) := max 1≤k≤m |c k − x| is the farthest distance function on the setof points c k . By Theorem 2.2, there exists a probability measure σ α such that U σ αα (x) =d α−Nm (x). Applying Tonelli’s Theorem, we havem∑∫∫inf U ν kα ≥ U σ αα (x)dν(x) = Uα(t)dσ ν α (t).Ek=1We estimate the potential Uα ν on R N . Let µ α be the α-equilibrium measure for E and letW α (E) be the α-energy for E. Let g(t) := U µ αα (t) − W α (E). By Frostman’s Theorem 2.1,we know g(t) ≤ 0 everywhere. On the other hand, Uα(t) ν − inf E Uα ν ≥ 0 for t ∈ E. Thusk=1U ν α(t) ≥ infEU α ν + U µ αα (t) − W α (E)on E. It follows by the Principle of Domination [15, Theorem 1.27 on p. 110 for α = 2 andTheorem 1.29 on p. 115 for 0 < α < 2] that this inequality holds in R N . Thus, noting thatσ α is a unit measure and again applying Tonelli’s Theorem, we findm∑∫inf U ν kα ≥ Uα(t)dσ ν α (t)E≥∫ (inf= infE U ν α += infE U ν α +∫U α ν +E∫∫)U µ α(t) − W α (E) dσ α (t)Uασα(x)dµ α (x) − W α (E)d α−Nm (x)dµ α (x) − W α (E).15

By minimizing over all m-tuples c k , we conclude thatwherem∑k=1infE∫C E (α, m) := minc k ∈EU ν k≥ C E (α, m) + infEm∑U ν k,k=1min |x − c k| 2−N dµ α (x) − W α (E).1≤k≤mWe now show C E (α, m) is the largest possible constant for a fixed m. We present twoproofs of this fact. We begin with the shorter one which requires E to be regular in thesense that U µ α∫ α (x) = W α (E) for all x ∈ E. Choose a set c ∗ k , k = 1, . . . , m, such thatdα−Nm (x)dµ α (x) attains its minimum on E m . Let d ∗ m(x) := min 1≤k≤m |x−c ∗ k | and iterativelydefine the setsIt is clear thatS 1 ={x ∈ supp(µ α ) : |x − c ∗ 1| = d ∗ m(x)},S k ={x ∈ supp(µ α ) \ ∪ k−1j=1 S j : |x − c ∗ k| = d ∗ m(x)}, k = 2, . . . , m.supp(µ α ) = ∪ m k=1S k and S k ∩ S j = ∅, k ≠ j.Hence we can decompose µ α along the sets S k such thatν ∗ k := µ α | Sk and µ α =m∑νk.∗k=1If E is regular, then ∫ |x − t| α−N dµ α (t) = W α (E) for each x ∈ E by Frostman’s Theorem.Applying this fact, along with Tonelli’s Theorem, we obtainm∑k=1infEU ν∗ kα≤==m∑k=1m∑∫k=1k=1U ν∗ kα (c ∗ k)|c ∗ k − x| α−N dν ∗ k(x)m∑∫(d ∗ m(x)) α−N dνk(x)∗∫= (d ∗ m(x)) α−N dµ α (x)∫∫= (d ∗ m(x)) α−N dµ α (x) − W α (E) + inft∈Em∑= C E (α, m) + inf U ν∗ jα .Ek=1|x − t| α−N dµ α (x)16

Hence C E (α, m) is sharp. The alternative proof uses points minimizing the discrete α-energy and does not require that E be regular. Let F n = {ξ l,n } n l=1 be the points of Ewhich minimize the discrete α-energy. We will break the set F n up using the points c ∗ k justas we broke up supp(µ α ) previously. Let F k,n be a subset of F n such that ξ l,n ∈ F l,n ifd ∗ m(ξ l,n ) = |ξ l,n − c ∗ k |, 1 ≤ l ≤ n. If there is overlap between the sets, assign ξ l,n to only oneset F k,n . It is clear that for any n ∈ N,Define the measuresso that for their potentialswe haveinfEF n = ∪ m k=1F k,n and F k,n ∩ F j,n = ∅, k ≠ j.νk,n ∗ = 1 ∑δ ξl,n ,nξ l,n ∈F k,np ∗ k,n(x) = 1 ∑|x − ξ l,n | α−N , k = 1, . . . , m,nξ l,n ∈F k,np∗ k,n(x) ≤ 1 ∑|c ∗ k − ξ l,n | α−N = 1 ∑(d ∗nnm(ξ l,n )) α−N .ξ l,n ∈F k,n ξ l,n ∈F k,nIt follows from the weak ∗ convergence of ν n := ∑ mk=1 ν∗ k,n = ∑ 1 nn l=1 δ ξ l,nto µ α , as n → ∞,thatlim supn→∞m∑k=1infE1n∑p∗ k,n(x) ≤ lim (d ∗n→∞ nm(ξ k,n )) α−Nk=1∫= (d ∗ m(x)) α−N dµ α (x).Applying Lemma 4.1 to the potential p ∗ n of ν ∗ n we find thatIt follows thatlim supn→∞m∑k=1lim infn→∞ Ep∗ n = W α (E).infEp∗ n ≤ C E (α, m) + limn→∞infE p∗ n.Hence we have asymptotic equality in (9) as n → ∞ with m ≥ 2 being fixed, which showsthat C E (α, m) is the largest possible constant for each m. Since d m ≤ d E everywhere, wehave C E (α, m) ≥ C E (α).Proof of Corollary 2.4. If m < card(D E ), then there is an x 0 ∈ supp(µ α ) such that d ∗ m(x 0 ) ∫ d α−NE(x)dµ α (x) and hence C E (α, m) > C E (α). Thisargument shows that if D E is infinite, then C E (α, m) > C E (α) for m ≥ 2. If m ≥ card(D E )17

then we may choose the points c ∗ k to include D E and hence d ∗ m(x) = d E (x) for x ∈ supp(µ α ).Thus C E (α, m) = C E (α).Let c k , k = 1, . . . , m, be a set of points in E that minimize the integral in the expressionof C E (α, m). Choose a point c m+1 ∈ ∂E. Then∫C E (α, m) = min |x − c k| α−N dµ α (x) − W α (E)1≤k≤m∫≥ min |x − c k| α−N dµ α (x) − W α (E)1≤k≤m+1≥ C E (α, m + 1).Hence the constants C E (α, m) are decreasing. It remains to show that their limit is C E (α).Let {a k } ∞ k=1 be a countable dense subset of E. Then∫C E (α) ≤ C E (α, m) ≤ min |x − a k| α−N dµ α (x) − W α (E).1≤k≤mFurther, applying the Dominated Convergence Theorem, we have∫∫lim min |x − a k| α−N dµ α (x) = lim min |x − a k| α−N dµ α (x)m→∞ 1≤k≤m m→∞ 1≤k≤m∫= d α−NE(x)dµ α (x).The result follows.Proof of Corollary 2.5. We show the minimal dominant set is infinite and then the resultfollows from Corollary 2.4. Suppose to the contrary that D E = {x j } s j=1 is finite. Let J ⊂ ∂Ebe a single connected component of the boundary. DefineJ k := {x ∈ J : d E (x) = |x − x k |}, k = 1, . . . , s.For each x ∈ J k , the segment [x, x k ] is orthogonal to ∂E at x k , by the smoothness assumption.Hence, each J k is contained in the normal line to ∂E at x k , k = 1, . . . , s. We thus obtainthat J = ∪ s k=1 J k is contained in a union of straight lines which is a contradiction.Proof of Example 2.6. To calculate the quantity min ck ∈T∫min1≤k≤m |x − c k | α−N dµ α (x), wefollow an idea of Boyd [8]. Let c k = −e iψ k , k = 1, . . . , m, with ψk < ψ k+1 , and for notationalconvenience let ψ 0 = ψ m . Then we have max 1≤k≤m |e iθ − c k | = |e iθ + e iψ k | = |ei(θ−ψ k ) + 1| for18

ψ k−1 +ψ k2≤ θ ≤ ψ k+ψ k+12and hence∫min |x − c k| α−N dµ α (x) = 11≤k≤m 2π= 1 πm∑∫ ψ k +ψ k+12ψ k−1 +ψ kk=1 2∫ ψ k −ψ k−12m∑k=1= 2α−2π=2 α−2 2 π0m∑k=1∫ ψ k −ψ k−120m∑I(θ k ),k=1|e i(θ−ψ k) + 1| α−2 dθ|e iθ + 1| α−2 dθcos α−2 ( θ2)dθwhere θ k = ψ k−ψ k−1and I(θ4 k ) = ∫ θ kcos α−2 (θ) dθ. Since I(θ0 k ) is strictly convex for 0 < θ k

[3] A. Baernstein, II, R. S. Laugesen, and I. E. Pritsker, Moment inequalities for equilibriummeasures in the plane, Pure Appl. Math. Q. 7 (2011), 51–86.[4] S. V. Borodachov and N. Bosuwan, Asymptotics of discrete Riesz d-polarization onsubsets of d-dimensional manifolds, Potential Anal. (to appear).[5] S. V. Borodachov, D.P. Hardin and E.B. Saff, Asymptotics of discrete Riesz polarizationfor rectifiable sets, (manuscript), 2013.[6] P. B. Borwein, Exact inequalities for the norms of factors of polynomials, Canad. J.Math. 46 (1994), 687–698.[7] D. W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika39 (1992), 341–349.[8] D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc.26 (1994), 449–454.[9] T. Erdélyi and E. B. Saff, Riesz polarization inequalities in higher dimensions, J. Approx.Theory 171 (2013), 128–147.[10] B. Farkas and B. Nagy, Transfinite diameter, Chebyshev constant and energy onlocally compact spaces, Potential Anal. 28 (2008) 241–260.[11] A. O. Gel’fond, Transcendental and Algebraic Numbers, Dover Publications Inc., NewYork, 1960.[12] D. P. Hardin, A. Kendall and E. B. Saff, Polarization optimality of equally spaced pointson the circle for discrete potentials, Discrete & Comp. Geometry 50 (2013), 236–243.arXiv number: 1208.5261v1.[13] H. Kneser, Das maximum des produkts zweies polynome, Sitz. Preuss. Akad. Wiss.Phys.-Math. Kl. (1934) 429–431.[14] A. Kroó and I. E. Pritsker, A sharp version of Mahler’s inequality for products ofpolynomials, Bull. London Math. Soc. 31 (1999), 269–278.[15] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York,1972.[16] K. Mahler, An application of Jensen’s formula to polynomials, Mathematika 7 (1960),98–100.[17] M. Ohtsuka, On various definitions of capacity and related notions, Nagoya Math. J.30 (1967), 121–127.[18] I. E. Pritsker, Products of polynomials in uniform norms, Trans. Amer. Math. Soc. 353(2001), 3971–3993.20

[19] I. E. Pritsker, Inequalities for sums of Green potentials and Blaschke products, Bull.Lond. Math. Soc. 43 (2011), 561–575.[20] I. E. Pritsker and S. Ruscheweyh, Inequalities for products of polynomials. I, Math.Scand. 104 (2009), 147–160.[21] I. E. Pritsker and S. Ruscheweyh, Inequalities for products of polynomials. II, AequationesMath. 77 (2009), 119–132.[22] I. E. Pritsker and E. B. Saff, Reverse triangle inequalities for potentials, J. Approx.Theory 159 (2009), 109–127.[23] W. Wise, Potential theory and geometry of the farthest distance function, PotentialAnal. (2013) published online.I. E. PritskerDepartment of Mathematics, Oklahoma State University, 401 Mathematical Sciences,Stillwater, OK 74078-1058, USAigor@math.okstate.eduE. B. SaffCenter for Constructive Approximation, Department of Mathematics, Vanderbilt University,Nashville, TN 37240, USAedward.b.saff@vanderbilt.eduW. WiseDepartment of Mathematics, Oklahoma State University, 401 Mathematical Sciences,Stillwater, OK 74078-1058, USAninadawn@gmail.com21