Kotrbaty's theorem on valuations and geometric inequalities for convex bodies
aa r X i v : . [ m a t h . M G ] O c t Kotrbat´y’s theorem on valuations andgeometric inequalities for convex bodies.
Semyon Alesker ∗ Department of Mathematics, Tel Aviv University, Ramat Aviv69978 Tel Aviv, Israele-mail: [email protected]
Abstract
Very recently J. Kotrbat´y [13] has proven general inequalities fortranslation invariant smooth valuations formally analogous to the Hodge-Riemann bilinear relations in the K¨ahler geometry. The goal of thisnote is to prove several inequalities for mixed volumes of convex bodiesusing Kotrbat´y’s theorem as the main tool. Very recently J. Kotrbat´y [13] has proven general inequalities for transla-tion invariant smooth valuations analogous to the Hodge-Riemann bilinearrelations from the K¨ahler geometry, see Theorem 2.14 below. The goal ofthis note is prove several new inequalities for mixed volumes of convexbodies using his theorem as the main tool.To formulate the main results let us introduce a notation. Let R n de-note the standard Euclidean space of dimension n . Let us denote by ι , ι , ∆ : R n −→ R n = R n × R n the maps ι ( x ) = ( x, , ι ( x ) = (0 , x ) , ∆( x ) = ( x, x ) . We refer to the book [16] for the notions of mixed volumes and mixed areameasures. The main result of this paper is ∗ Partially supported by ISF grant 865/16 and the US - Israel BSF grant 2018115. .1 Theorem. Let n ≥ . Let A , . . . , A n − ⊂ R n be convex compactsubsets. Let B ⊂ R n be the unit Euclidean ball.(1) (a) Then one has inequality for mixed volumes in R n = R n × R n V ( ι A , . . . , ι A n − ; ι A , . . . , ι A n − ; ∆( B )[2]) ≥ V ( ι A , . . . , ι A n − ; − ι A , . . . , − ι A n − ; ∆( B )[2]) . (b) In the special case when A = · · · = A n − =: A with A having non-empty interior and smooth boundary with positive Gauss curvature theequality in the above inequality is achieved if and only if A has a center ofsymmetry.(2) (a) Furthermore V ( ι A , . . . , ι A n − ; ι A , . . . , ι A n − ; ∆( B )[2]) ++ V ( ι A , . . . , ι A n − ; − ι A , . . . , − ι A n − ; ∆( B )[2]) ≤ γ n V ( A , . . . , A n − , B ) , where γ n is a constant depending on n only and uniquely characterized bythe property that if A = · · · = A n − = B then in the above inequalitythere is an equality.(b) In the special case when A = · · · = A n − =: A with A being centrallysymmetric, having non-empty interior and smooth boundary with positiveGauss curvature the equality in the above inequality is achieved if and onlyif A is a Euclidean ball of arbitrary radius.(3) Let n ≥ . Let K , K , L , L ⊂ R n be centrally symmetric convexcompact sets with smooth boundary with positive Gauss curvature. Let usassume the following equality of the mixed area measures: S ( K , K , B [ n − , · ) = S ( L , L , B [ n − , · ) . (1.1) Then one has V ( K [2] , K [2] , B [ n − V ( L [2] , L [2] , B [ n − ≥ V ( K , K , L , L , B [ n − . (1.2) (1) It is expected that in part (3) the assumption that allbodies K i , L i are centrally symmetric is unnecessary. That would followfrom Kotrbat´y’s conjecture 2.13.(2) If in part (3) of the theorem one takes K = K =: K and L = L =: L then the inequality (1.2) is weaker than what is actually known: in this case2he equality is achieved. Indeed the Alexandrov-Fenchel-Jessen theorem(see [16], Corollary 8.1.4) says that the assumption (1.1) implies that K and L have to be translates of each other.(3) In part (2)(b) the assumption of central symmetry of A cannot beomitted at least in the case of n = 2. One can show that for n = 2 theinequality (2)(a) is equivalent to the standard isoperimetric inequality forthe body A − A : area ( A − A ) ≤ length ( ∂ ( A − A )) π = length ( ∂A ) π . The equality here is achieved if and only if A − A is a Euclidean disk ofarbitrary radius. Valuations on convex sets are finitely additive functionals on convex com-pact subsets in R n (the Definition 2.1). It is a classical object of convexity.During the last 25 years there was a considerable progress in the theory.There was a breakthrough in 1995 when the Klain-Schneider characteri-zation of translations invariant simple continuous valuations was obtained[12], [15] which opened a way to further progress. In particular the Mc-Mullen’s conjecture has been solved [1]. Subsequent progress led to adiscovery of a few new structures on valuations, to posing new questions,and to opening new directions of the research. Some of these structuresturned out to be useful in integral geometry, see e.g. [8], [10]. Examples ofnew structures relevant to this paper are the product on smooth valuationsintroduced by the author [3] and the convolution on them introduced byBernig and Fu [9] (see Theorems 2.7 and 2.9 below).The most basic examples of continuous translation invariant valuations aregiven by mixed volumes of convex sets. Mixed volume is a well studiedclassical object in convexity of its own right. One of the most remarkableproperties of the mixed volume is the Alexandrov-Fenchel inequality (see[16], Theorem 7.3.1). Kotrbat´y [13] has stated a conjecture, which is ageneralized version of a part of his Conjecture 2.13 below, which impliesthe Alexandrov-Fenchel inequality. Furthermore he deduced there a specialcase of the latter inequality from his Theorem 2.14 below.The main result of this paper, Theorem 1.1, shows that Kotrbat´y’s work[13] can also be used to obtain new geometric inequalities using valuations This observation is a result of a correspondence with T. Wannerer. In Section 2 we review some background from the valuations theory nec-essary to formulate the Kotrbat´y’s theorem. Section 3 contains proofs ofthe main results. Theorem 1.1 summarizes Theorems 3.4, 3.5, and 3.7 inthe main text.
I thank J. Kotrbat´y for useful remarks on the firstversion of the paper. I thank R. Schneider and T. Wannerer for usefulcorrespondences. In this section we summarize basic definitions and results on valuations.For general background on convexity and, in particular, mixed volumes werefer to [16].For an n -dimensional real vector space W we denote by K ( W ) the familyof all convex compact non-empty subsets of W . Being equipped withthe Hausdorff metric K ( W ) is a locally compact space by the Blaschkeselection theorem ([16], Theorem 1.8.7). A valuation is a map φ : K ( W ) −→ C which satisfies thefollowing additivity property: φ ( A ∪ B ) = φ ( A ) + φ ( B ) − φ ( A ∩ B )whenever A, B, A ∪ B ∈ K ( W ). A valuation φ is called continuous if it is continuous inthe Hausdorff metric. .3 Definition. A valuation φ is called translation invariant if φ ( K + x ) = φ ( K )for any K ∈ K ( W ) , x ∈ W .We will denote by V al ( W ) the space of continuous translation invariantvaluations on K ( W ). Being equipped with the topology of uniform con-vergence on compact subsets of K ( W ), V al ( W ) is a Banach space (see e.g.Lemma 7.0.3 in [6]). A valuation φ ∈ V al ( W ) is called i -homogeneous if φ ( λK ) = λ i φ ( K ) for any λ > , K ∈ K ( W ) . Let
V al i ( W ) ⊂ V al ( W ) denote the subspace of i -homogeneous valua-tions; clearly it is a closed linear subspace. The following result is calledMcMullen’s decomposition theorem. [McMullen [14]] V al ( W ) = ⊕ dim Wi =0 V al i ( W ) , where the sum runs over all integers between 0 and dim W . It is easy to see that 0-homogeneous valuations
V al ( W ) is a 1-dimensionalspace spanned by the Euler characteristic χ . V al dim W ( W ) is also 1-dimensional and is spanned by a Lebesgue measure; this result is dueto Hadwiger [11] (see also [6], Theorem 3.1.1). The group GL ( W ) of invertible linear transformations acts linearly andcontinuously on V al ( W ) as( gφ )( K ) = φ ( g − K ) for any g ∈ GL ( W ) , φ ∈ V al ( W ) , K ∈ K ( W ) . A valuation φ ∈ V al ( W ) is called smooth if the map GL ( W ) −→ V al ( W ) given by g gφ is C ∞ -differentiable. By definition χ takes value 1 on each non-empty convex compact set. V al ∞ ( W ). A well knownrepresentation theoretical result is that V al ∞ ( W ) is a GL ( W )-invariantlinear subspace dense in V al ( W ). Moreover V al ∞ ( W ) admits a canonicalFr´echet topology, called the Garding topology, which is stronger than thatinduced from V al ( W ). We do not need the precise definition of it, but itis important to know that the product and the convolutions on V al ∞ ( W )discussed below are continuous with respect to Garding topology. Clearly V al ∞ ( W ) also satisfies the McMullen’s decomposition theorem.An important example of smooth valuations which will be used below is K vol ( K + A ) where A ∈ K ( W ) has infinitely smooth boundary andpositive Gauss curvature (i.e. all principal curvatures of the boundary arestrictly positive; this class of convex bodies is independent of any Euclideanmetric). Here K + A is the Minkowski sum K + A := { k + a | k ∈ K, a ∈ A } . Furthermore the mixed volume K V ( K [ i ] , A , . . . , A n − i ) is a smoothvaluation provided A , . . . A n − i ∈ K ( W ) have smooth boundary and pos-itive Gauss curvature. Here and below the notation K [ i ] means that thebody K is repeated i times. Let us discuss the product on valuations. We denote by ι , ι , ∆ : W −→ W × W the imbeddings given by ι ( w ) = ( w, , ι ( w ) = (0 , w ) , ∆( w ) =( w, w ). [[3]] (1) There exists a continuous (in the Garding topol-ogy) bilinear map called product V al ∞ ( W ) × V al ∞ ( W ) −→ V al ∞ ( W ) which is uniquely characterized by the following property: Let A, B ∈K ( W ) have smooth boundary and positive curvature. Consider smoothvaluations φ A ( • ) = vol W ( • + A ) , φ B ( • ) = vol W ( • + B ) where vol W is aLebesgue measure on W . Then their product is ( φ A · φ B )( K ) = vol W (∆( K ) + ( A × B )) , where vol W = vol W × vol W is the product measure.(2) Equipped with this product V al ∞ ( W ) is an associative commutativealgebra with the unit χ (the Euler characteristic). V al ∞ ( W ) is a graded algebra with respect to the McMullen’s decompo-sition, i.e. V al ∞ i ( W ) · V al ∞ j ( W ) ⊂ V al ∞ i + j ( W ) . (4) (Poincar`e duality) Consider the product map of smooth valuations ofcomplementary degree of homogeneity V al ∞ i ( W ) × V al ∞ dim W − i ( W ) −→ V al ∞ dim W ( W ) = C · vol W . This is a perfect pairing, i.e. for any = φ ∈ V al ∞ i ( W ) there exists ψ ∈ V al ∞ dim W − i ( W ) such that φ · ψ = 0 . Let φ ( • ) = V ( • [ i ] , A , . . . , A n − i ) , ψ ( • ) = V ( • [ j ] , B , . . . , B n − j )where i + j ≤ n and A p , B q ∈ K ( W ) have smooth boundary with positiveGauss curvature.(1) Then( φ · ψ )( • ) = (cid:18) nn (cid:19)(cid:18) i + ji (cid:19) − V (∆( • )[ i + j ]; ι A , . . . , ι A n − i ; ι B , . . . , ι B n − j ) , (2.1)where the mixed volume in the right hand side is taken in W × W .(2) If i + j = n then the expression for the product can be presented in amore explicit form by Proposition 2.2 in [3]:( φ · ψ )( • ) = (cid:18) ni (cid:19) − V ( A , . . . , A n − i , − B , . . . , − B i ) vol W ( • ) , (2.2)where the mixed volume is taken in W . Let us discuss convolution on valuations which was introduced by Bernigand Fu. [[9]] Let us fix a positive Lebesgue measure vol W on thespace W .(1) There exists a continuous (in the Garding topology) bilinear map calledconvolution V al ∞ ( W ) × V al ∞ ( W ) −→ V al ∞ ( W ) The convolution does depend on a choice of vol W . hich is uniquely characterized by the following property: Let A, B ∈K ( W ) have smooth boundary and positive Gauss curvature. Considersmooth valuations φ A ( • ) = vol W ( • + A ) , φ B ( • ) = vol W ( • + B ) . Then ( φ A ∗ φ B )( • ) = vol W ( • + A + B ) . (2) Equipped with the convolution V al ∞ ( W ) is an associative commutativealgebra with unit vol W .(3) V al ∞ i ( W ) ∗ V al ∞ j ( W ) ⊂ V al ∞ i + j − dim W ( W ) . It was shown in [5] that the topological algebra
V al ∞ ( W ) equipped withthe product is isomorphic to the topological algebra V al ∞ ( W ∗ ) equippedwith convolution ; moreover an isomorphism may be chosen to commutewith the natural action of the subgroup of GL ( W ) of volume preservingtransformations. Let φ ( • ) = V ( • [ i ] , A , . . . , A n − i ) , ψ ( • ) = V ( • [ j ] , B , . . . , B n − j )where i + j ≥ n and A p , B q ∈ K ( W ) have smooth boundary with positiveGauss curvature. Then( φ ∗ ψ )( • ) = (cid:18) i + jn (cid:19)(cid:18) i + ji (cid:19) − V ( • [ i + j − n ]; A , . . . , A n − i ; B , . . . , B n − j ) , (2.3)where the mixed volume in the right hand side is in W . Let us discuss hard Lefschetz type theorems on valuations. There aretwo versions: for the product and for the convolution. We denote by V i , i = 0 , , . . . , n , the i th intrinsic volume. Recall that V i ( • ) is proportionalwith positive coefficient to the mixed volume with the unit Euclidean ball V ( • [ i ] , B [ n − i ]). The product V i · V j is proportional with positive coefficientto V i + j , and the convolution V i ∗ V j is proportional with positive coefficientto V i + j − n . Let us denote n = dim W . Let ≤ i < n/ .(1) The linear map V al ∞ i ( W ) −→ V al ∞ n − i ( W ) given by φ φ · ( V ) n − i isan isomorphism.(2) The linear map V al ∞ n − i ( W ) −→ V al ∞ i ( W ) given by ψ ψ ∗ ( V n − ) ∗ ( n − i ) is an isomorphism. W ∗ denotes the dual space of W . Let us state Kotrbat´y’s results. There will be two equivalent versionsformulated either in terms of product or convolution.
Let i ≤ n/ n = dim W .(1) A valuation φ ∈ V al ∞ i ( W ) is called primitive if φ · ( V ) n − i +1 = 0 . (2) A valuation ψ ∈ V al ∞ n − i ( W ) is called co-primitive if ψ ∗ ( V n − ) ∗ ( n − i +1) =0.For i ≤ n/ Q on V al ∞ i ( W ) with valuesin V al n ( W ) = C · vol W by Q ( φ ) := ( − i φ · ¯ φ · ( V ) n − i and the Hermitian form ˜ Q on V al ∞ n − i ( W ) with values in V al ( W ) = C · χ by ˜ Q ( ψ ) = ( − i ψ ∗ ¯ ψ ∗ V ∗ ( n − i ) n − . Kotrbat´y has formulated the following conjecture which is an analogue ofthe Hodge-Riemann bilinear relations from K¨ahler geometry. [[13]] Let i ≤ n/ .(1) Let φ ∈ V al ∞ i ( W ) be a non-zero primitive even (resp. odd) valuation.Then Q ( φ ) > (resp. Q ( φ ) < ).(2) Let ψ ∈ V al ∞ n − i ( W ) be a non-zero co-primitive valuation. Then ˜ Q ( ψ ) > . Kotrbat´y has shown in [13] that parts (1) and (2) of the conjecture areequivalent. He proved the conjecture in the following special cases. [[13]] Conjecture 2.13 holds for even valuations for any i as in the conjecture, and for odd valuations for i = 0 , . This theorem will be applied to the main results of this paper for valua-tions given by mixed volumes. Their connection to the product and theconvolution is given in Examples 2.8 and 2.10.9
Proof of the main results.
Let W be an n -dimensional vector space with a fixed positive Lebesgue mea-sure vol W . We denote the linear maps∆ : W ֒ → W given by ∆ ( w ) = ( w, w ) , ∆ : W ֒ → W given by ∆ ( w ) = ( w, w, w ) . As previously we denote the imbeddings ι , ι : W ֒ → W × W by ι ( w ) =( w, ι ( w ) = (0 , w ). In a similar way, by the abuse of notation, we willdenote analogous imbeddings ι , ι , ι : W ֒ → W as imbeddings of W intothe corresponding copy of W in the triple product W , Let
A, B, C ⊂ W be convex compact subsets with smooth bound-ary and positive Gauss curvature. Consider smooth valuations φ A ( • ) := vol W ( • + A ) , and similarly for B, C . Then ( φ A · φ B · φ C )( K ) = vol W (∆ ( K ) + ( A × B × C )) for any K ∈ K ( W ) , where vol W is the product measure of vol W taken 3 times. Analogous statement can be proven not only for a triple prod-uct of valuations of the above form, but for any number of them.
Proof.
First let us show that for any ψ ∈ V al ∞ ( W ) and φ C as in thelemma one has ( ψ · φ C )( K ) = Z W ψ ( K ∩ ( y − C )) dy. (3.1)10or by continuity and by solution of the McMullen’s conjecture [ ? ] it sufficesto assume that ψ is of the form φ A . Then by Fubini theorem( φ A · ψ C )( K ) = vol W (∆ ( K ) + ( A × C )) = Z W vol W ([∆ ( K ) + ( A × C )] ∩ ( W × { y } )) dy = Z W vol W (([∆ ( K ) + (0 × C )] ∩ ( W × { y } )) + A ) dy = Z W φ A ([∆ ( K ) + (0 × C )] ∩ ( W × { y } )) dy = Z W φ A ( { k | k ∈ K and ∃ c ∈ C s.t. k + c = y } ) dy = Z W φ A ( K ∩ ( y − C )) dy which is (3.1).Now let us consider vol W (∆ ( K ) + ( A × B × C )) = Z W vol W (cid:0)(cid:0) [∆ ( K ) + (0 × × C )] ∩ ( W × { y } ) (cid:1) + ( A × B × (cid:1) dy = Z vol W ( { ( k, k ) | k ∈ K and ∃ c ∈ C s.t. k + c = y } + ( A × B )) dy = Z vol W (∆ ( K ∩ ( y − C )) + ( A × B )) dy = Z ( φ A · φ B )( K ∩ ( y − C )) dy ( . ) = (( φ A · φ B ) · φ C )( K ) . Q.E.D.
Let us given two n − -tuples of convex compact sets A i , B i , i =1 , . . . , n − , in n -dimensional space W , and two more convex compact sets C , C . Let us assume that all the bodies have smooth boundary and positivecurvature. Consider valuations α ( • ) = V ( • , A , . . . , A n − ) ,β ( • ) = V ( • , B , . . . , B n − ) ,γ ( • ) = V ( • [ n − , C , C ) . hen their product is equal to α · β · γ = c n V ( ι A , . . . , ι A n − , ι B , . . . , ι B n − , − ∆ ( C ) , − ∆ ( C )) · vol W , where c n > is a constant depending on n only. Proof . Clearly α ( • ) = 1 n ! ∂∂λ . . . ∂∂λ n − vol W ( • + n − X i =1 λ i A i ) ,β ( • ) = 1 n ! ∂∂µ . . . ∂∂µ n − vol W ( • + n − X j =1 µ j B j ) ,γ ( • ) = (cid:18) n (cid:19) − ∂∂ε ∂∂ε vol W ( • + ε C + ε C ) , where all the partial derivatives are taken at 0. Hence by Lemma 3.1 we get( α · β · γ )( K ) = c ′ n V (∆ ( K )[ n ]; ι A , . . . , ι A n − ; ι B , . . . , ι B n − ; ι C , ι C ) , (3.2)where K ∈ K ( W ) is arbitrary, and c ′ n is a constant depending on n only. Theproduct is n -homogenous valuation in K and hence has to be proportionalto vol W ( K ) by the Hadwiger theorem [11]. However this fact can be seenmore directly using Theorem 5.3.1 in [16] which will also give the requiredcoefficient of proportionality.The latter result says that if E ⊂ X be a k -dimensional linear subspaceof an N -dimensional vector space X . Let π : X −→ X/E be the quotientmap. Let L , . . . L N − k ∈ K ( X ). Then for any K ⊂ EV ( K [ k ] , L , . . . , L N − k ) = c ′′ vol E ( K ) V ( πL , . . . , πL N − k ) , where vol E is a Lebegue measure on E , V in the left (resp. right) hand sideis the mixed volume with respect to a Lebesgue measure on X (resp. X/E ),and c ′′ > E, X, X/E , itsprecise value is not important for us.We apply this result for E = ∆ ( W ) ⊂ X = W . Then this theorem and(3.2) imply( α · β · γ )( K ) = c ′′ n V ( πι A , . . . πι A n − ; πι B , . . . πι B n − ; πι C , πι C ) · vol W ( K ) , (3.3) c n can be computed explicitly, but its value is not important in this paper. π : W −→ W / ∆ ( W ) is the canonical quotient map. There exists aunique isomorphism of vector spaces Q : W / ∆ ( W ) ˜ −→ W , given by ( Q ◦ π )( w , w , w ) = ( w − w , w − w ). Applying Q to all convexbodies A i , B i , C i and taking into account that ( Qπι )( A ) = ι A, ( Qπι )( B ) = ι B, ( Qπι )( C ) = − ∆ ( C ), one gets ( α · β · γ )( K ) = c n V (( Qπι ) A , . . . ( Qπι ) A n − ; ( Qπι ) B , . . . ( Qπι ) B n − ; ( Qπι ) C , ( Qπι ) C ) · vol W ( K ) = c n V ( ι A , . . . , ι A n − ; ι B , . . . , ι B n − ; − ∆ ( C ) , − ∆ ( C )) · vol W ( K ) , where the last mixed volume is in W . Q.E.D. (a) Let A , . . . A n − ⊂ R n be convex compact subsets of theEuclidean space. Let B ⊂ R n be the unit Euclidean ball. Then one hasinequality for mixed volumes in R n = R n × R n V ( ι A , . . . , ι A n − ; ι A , . . . , ι A n − ; ∆ ( B )[2]) ≥ V ( ι A , . . . , ι A n − ; − ι A , . . . , − ι A n − ; ∆ ( B )[2]) . (b) In the special case when A = · · · = A n − =: A with A having non-empty interior and smooth boundary with positive Gauss curvature the equal-ity in the above inequality is achieved if and only if A has a center of sym-metry. Proof.
We may and will assume that all bodies A i have smooth boundaryand positive curvature. Consider the valuation φ ( • ) = V ( • , A , . . . , A n − ) − V ( • , − A , . . . , − A n − ) .φ is a 1-homogeneous odd valuation, and hence it is primitive. Hence byKotrbat´y’s theorem 2.14 φ · V ( • [ n − , B [2]) ≥ . (3.4)Computing this product explicitly using Lemma 3.3 one gets the result (a).Note that if φ = 0 then there is a strict inequality in (3.4). Hence if thereis an equality in the inequality of part (a) of the theorem it follows that13 = 0. In the case A = · · · = A n − = A that is equivalent to the equality ofthe area measures S n − ( A, · ) = S n − ( − A, · ) . By Theorem 8.1.1 in [16] the latter condition is satisfied if and only if A and − A are translates of each other. This implies part (b) of the theorem.Q.E.D. Let n ≥ . Let A , . . . A n − ⊂ R n be convex compact subsetsof Euclidean space. Let B ⊂ R n be the unit Euclidean ball.(a) One has inequality for mixed volumes in R n = R n × R n in the lefthand side and in R n in the right hand side V ( ι A , . . . , ι A n − ; ι A , . . . , ι A n − ; ∆ ( B )[2]) ++ V ( ι A , . . . , ι A n − ; − ι A , . . . , − ι A n − ; ∆ ( B )[2]) ≤ γ n V ( A , . . . , A n − , B ) , where γ n is a constant depending on n only and uniquely characterized by theproperty that if A = · · · = A n − are equal to B then in the above inequalitythere is an equality.(b) In the special case when A = · · · = A n − =: A with A being a centrallysymmetric convex body with non-empty interior and smooth boundary withpositive Gauss curvature, the equality in the above inequality is achieved ifand only if A is a Euclidean ball of arbitrary radius. Proof.
We may and will assume that all convex bodies A i have smoothboundary with positive curvature. Consider the valuation φ ( • ) := V ( • , A , . . . , A n − ) + V ( • , − A , . . . , − A n − ) , Clearly φ is an even 1-homogeneous valuation. We are going to show thatthere exists λ ∈ R such that the valuation ψ ( • ) := φ ( • ) − λV ( • , B [ n − ψ · V ( • [ n − , B ) , or equivalently φ · V ( • [ n − , B ) = λV ( • , B [ n − · V ( • [ n − , B ) . (3.5)By (2.2) this is equivalent to2 V ( B, A , . . . , A n − ) = λvol ( B ) . λ = 2 V ( B, A , . . . , A n − ) vol ( B ) . (3.6)For this particular λ let us use Kotrbat´y’s theorem 2.14:0 ≥ ψ · V ( • [ n − , B [2]) = φ V ( • [ n − , B [2]) + λ V ( • , B [ n − · V ( • [ n − , B [2]) −− λφV ( • , B [ n − · V ( • [ n − , B [2])By (3.5) the last summand is equal to − λ V ( • , B [ n − · V ( • [ n − , B [2]).Hence we obtain0 ≥ φ V ( • [ n − , B [2]) − λ V ( • , B [ n − · V ( • [ n − , B [2]) (3.7)By Lemma (3.3) φ V ( • [ n − , B [2]) = c n vol · ( V ( ι A , . . . , ι A n − , ι A , . . . , ι A n − , ∆ ( B )[2]) + V ( − ι A , . . . , − ι A n − , − ι A , . . . , − ι A n − , ∆ ( B )[2]) +2 V ( ι A , . . . , ι A n − , − ι A , . . . , − ι A n − , ∆ ( B )[2])) =2 c n vol · ( V ( ι A , . . . , ι A n − , ι A , . . . , ι A n − , ∆ ( B )[2]) + V ( ι A , . . . , ι A n − , − ι A , . . . , − ι A n − , ∆ ( B )[2])) . Similarly V ( • , B [ n − · V ( • [ n − , B [2]) = c n vol · V ( ι B [ n − , ι B [ n − , ∆ [2]) . Substituting the last two equalities into (3.7) we get V ( ι A , . . . , ι A n − , ι A , . . . , ι A n − , ∆ ( B )[2]) + V ( ι A , . . . , ι A n − , − ι A , . . . , − ι A n − , ∆ ( B )[2]) ≤ λ V ( ι B [ n − , ι B [ n − , ∆ ( B )[2]) ( . ) =( V ( B, A , . . . , A n − )) V ( ι B [ n − , ι B [ n − , ∆ ( B )[2]) vol ( B ) | {z } γ n . Thus part (a) of the theorem is proven.15o prove part (b) assume that A = − A and there is an equality in theinequality of part (a). This is equivalent to ψ = 0, or V ( • , A [ n − λ V ( • , B [ n − . In other words S n − ( A, · ) = λ S n − ( B, · ). By Theorem 8.1.1 in [16] A = λ B .Q.E.D.If we combine Theorems 3.4 and 3.5 we get Let n ≥ . Let A , . . . A n − ⊂ R n be convex compact subsetsof Euclidean space. Let B ⊂ R n be the unit Euclidean ball. Then one has V ( ι A , . . . , ι A n − ; − ι A , . . . , − ι A n − ; ∆ ( B )[2]) ≤ γ ′ n ( V ( A , . . . , A n − , B )) , where γ ′ n is a constant depending on n only and uniquely characterized by theproperty that if A = · · · = A n − = B then in the above inequality there isan equality. Let n ≥ . Let K , K , L , L ⊂ R n be centrally symmetricconvex compact sets with smooth boundary with positive Gauss curvature. Let B ⊂ R n denote the unit Euclidean ball. Let us assume the following equalityof the mixed area measures: S ( K , K , B [ n − , · ) = S ( L , L , B [ n − , · ) . (3.8) Then one has V ( K [2] , K [2] , B [ n − V ( L [2] , L [2] , B [ n − ≥ V ( K , K , L , L , B [ n − . Proof.
Let us consider
V al ∞ ( R n ) as algebra equipped with the Bernig-Fuconvolution. Consider the even smooth n − φ ( • ) = V ( • [ n − , K , K ) − V ( • [ n − , L , L ) . Let us show that it is primitive in the sense of convolution, i.e. φ ∗ V n − n − = 0,or equivalently φ ∗ V ( • [3] , B [ n − V ( • , K , K , B [ n − − V ( • , L , L , B [ n − . In fact γ ′ n = γ n / γ n is the constant from Theorem 3.5. κ n > ≤ φ ∗ V = κ n vol · ( V ( K [2] , K [2] , B [ n − V ( L [2] , L [2] , B [ n − − V ( K , K , L , L , B [ n − . This is exactly the statement of the theorem. Q.E.D.
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