aa r X i v : . [ m a t h . M G ] J u l MIXED L p PROJECTION INEQUALITY
ZHONGWEN TANG AND LIN SI ∗ Abstract.
In this paper, the mixed L p -surface area measures are defined and the mixed L p Minkowski inequality is obtained consequently. Furthermore, the mixed L p projectioninequality for mixed projection bodies is established.Key Word. mixed L p -surface area measures, mixed L p -projection body, mixed L p Minkowskiinequality, mixed L p projection inequalityAMS Classification: 52A40 Introduction
In the n -dimensional Euclidean space E n , mixed volumes, which are the generalizationof surface, volume, mean width and so on of a convex body(compact, convex subsets withnonempty interiors in E n ) are the core concept in the classical Brunn-Minkowski theory.Surface area measures are one of important geometric measures of convex bodies in E n forthe integral formula of mixed volumes. Usually, the surface area measure is interpreted asthe first variation of volume with respect to the Minkowski addition.The mixed volume V ( K, L ) of convex bodies K and L , admits the following integralrepresentation V ( K, L ) = 1 n lim ε → + V ( K + εL ) − V ( K ) ε = 1 n Z S n − h L dS K ( u ) , where h L is the support function of L , and S K ( u ) is the area measure of K . The famousMinkowski’s first inequality states that(1.1) V n ( K, L ) ≥ V n − ( K ) V ( L ) , with equality if and only if K and L are homothetic.In [3], Firey introduced the L p Minkowski combination (also known as the Minkowski-Fireycombination) for each p ≥
1. The L p Brunn-Minkowski theory, combined the L p Minkowskicombination and the volume as a generalization of the classical Brunn-Minkowski theoryhas attracted increasing interest in recent years (see, e.g., [2, 6, 7, 12–18, 24]). One of themost important concept in the L p Brunn-Minkowski theory (see, [12]) is the L p -surface areameasure which is a Borel measure (defined on the unit sphere S n − in E n ) for each convexbody in E n that contains the origin in its interior, and has a representation formula for the p -mixed volume due to Lutwak. The L p -surface area measure and its associated Minkowskiproblem in the L p Brunn-Minkowski theory were introduced in [12].Let K n denote the class of convex bodies in E n and K no denote the subset of K n thatcontains the origin as interiors. Supported by the Fundamental Research Funds for the Central Universities(2015ZCQ-LY-01, 2017ZY44). ∗ Corresponding author. ∗ If K, L ∈ K no and p ≥
1, then the L p -mixed volume V p ( K, L ) is defined by V p ( K, L ) = pn lim ε → + V ( K + p ε · L ) − V ( K ) ε = 1 n Z S n − h L dS p ( K, u ) , where S p ( K, · ) is the L p -surface area measure of K . The L p -mixed volume V p ( K, L ) is thecase i = 0 of the L p -mixed quermassintegral introduced by Lutwak in [12].The L p Minkowski inequality states that for p ≥ V np ( K, L ) ≥ V n − p ( K ) V p ( L ) , with equality if and only if K and L are dilates. Obviously, the case p = 1 of (1.2) is theMinkowski’s first inequality (1.1).If K, L ∈ K no , for p ≥ ≤ i ≤ n −
1, the L p -mixed quermassintegrals W p,i ( K, L ) isdefined by(1.3) W p,i ( K, L ) = pn − i lim ε → + W i ( K + p ε · L ) − W i ( K ) ε . The Minkowski inequality for the L p -mixed quermassintegrals states that for p ≥ ≤ i ≤ n − W n − ip,i ( K, L ) ≥ W n − i − pi ( K ) W pi ( L ) , with equality if and only if K and L are dilates.The purpose of this paper is to construct mixed L p -surface area measures and obtain themixed L p projection inequalities. Definition 1.
For
K, L ∈ K no , Q ∈ K n and ≤ t ≤ n − , p ≥ , the L p,t mixed volume isdefined by V p,t ( K, L, Q ) = pt + 1 lim ε → + V ( K + p ε · L, t + 1;
Q, n − t − − V ( K, t + 1;
Q, n − t − ε . Our first main result is the following L p -type Minkowski inequality for L p,t mixed volume. Theorem 1. If K, L ∈ K no , Q ∈ K n , then for p ≥ and ≤ t ≤ n − V np,t ( K, L, Q ) ≥ V t +1 − p ( K ) V p ( L ) V n − t − ( Q ) , with equality if and only if K , L are dilates and Q is dilate, up to translation. An important inequality involving the polar projection bodies was obtained by Petty [19](see [9] for an alternate proof). This inequality is now known as the Petty projection in-equality, i.e., for K ∈ K n , then V n − ( K ) V (Π ∗ K ) ≤ ω nn , with equality if and only if K is an ellipsoid. Here Π K is the projection body of K and Π ∗ K is the polar body of Π K .More generally, Lutwak (see, [9] or [10]) obtained the following mixed projection inequality,i.e., for K , · · · , K n − ∈ K n , then V ( K ) · · · V ( K n − ) V (Π ∗ ( K , · · · , K n − )) ≤ ω nn , with equality if and only if the K i are homothetic ellipsoids. IXED L p PROJECTION INEQUALITY 3
In [14], Lutwak, Yang and Zhang introduced L p -projection body Π p K and established the L p -Petty projection inequality, i.e., for K ∈ K no and 1 ≤ p < ∞ , V ( K ) ( n − p ) /p V (Π ∗ p K ) ≤ ω n/pn , with equality if and only if K is an ellipsoid centered at the origin. An alternative proof ofthe L p -Petty projection inequality was given by Campi and Grochi [2].In Section 2, we shall introduce the mixed L p -projection body Π p,t ( K, Q ), which is anextension of L p -projection body. The case t = n − L p -projection body is the L p -projection body Π p K . The second main result of this paper is the mixed L p -projectioninequalities, which is an extension of L p -Petty projection inequality. Theorem 2. If K ∈ K no , Q ∈ K n , then for p > and < t < n − , (1.6) V ( t +1 − p ) /p ( K ) V ( n − t − /p ( Q ) V (Π ∗ p,t ( K, Q )) ≤ ω n/pn , with equality if and only if K and Q are dilate ellipsoids centered at the origin. Preliminary results
For x ∈ E n \{ } , the support function of K ∈ K n is defined by(2.1) h K ( x ) = max { x · y : y ∈ K } , where x · y denotes the standard inner product of x and y in E n . For x ∈ E n and φ ∈ GL ( n ),the support function of the image φK = { φx : x ∈ K } is given by(2.2) h φK ( x ) = h K ( φ t x )where φ t denotes the transpose of φ . For K, L ∈ K n and s, t ≥
0, the Minkowski combination sK + tL is defined by sK + tL = { sx + ty : x ∈ K, y ∈ L } , or equivalently, h sK + tL = sh K + th L . For
K, L ∈ K no and s, t ≥
0, the L p Minkowski combination, s · K + p t · L , for each p ≥ h ps · K + p t · L = sh pK + th pL . The mixed volume of K , · · · , K n ∈ K n is denoted by V ( K , · · · , K n ) and is uniquelydetermined by the requirement that it be symmetric in its arguments. If K, L, Q ∈ K n , then V ( K, s ; L, t ; Q, n − s − t ) will be used to denote the mixed volume V ( K, · · · , K, L, · · · , L, Q, · · · , Q ) , in which K appears s times, L appears t times and Q appears n − s − t times.For K ∈ K no , one has K + p ε · K = (1 + ε ) p K by the definition of the L p Minkowskicombination. Hence, from the Definition 1, we can get that V p,t ( K, K, Q ) = V ( K, t + 1;
Q, n − t − V p,t ( K, K, K ) = V ( K )for all p ≥
1, 0 ≤ t ≤ n − ZHONGWEN TANG AND LIN SI ∗ The mixed area measure S ( K , · · · , K n − ; · ) associated with the convex bodies K , · · · , K n − in E n is a unique (positive) Borel measure on S n − , with the property that for a convex body K in E n , one has the integral representation V ( K , · · · , K n − , K ) = 1 n Z S n − h K ( u ) dS ( K , · · · , K n − ; u ) . If K, Q ∈ K n , then S ( K, t ; Q, n − t − · ) will be used to denote the mixed surface areameasures S ( K, · · · , K, Q · · · , Q ; · ) , in which K appears t times and Q appears n − t − S ( K, · · · , K ; · )= S K ( · ).If K ∈ K n and u ∈ S n − , let K u denote the orthogonal projection of K onto the 1-codimensional space u ⊥ perpendicular to u . The ( n − υ ( K u ), also called the brightness of K in the direction u . If K , · · · , K n − ∈ K n and u ∈ S n − , then the mixed volume of K u , · · · , K un − in u ⊥ is written as υ ( K u , · · · , K un − )and called the mixed brightness of K , · · · .K n − in the direction u .The projection body Π K of K ∈ K n is an origin-symmetric convex body defined by h Π K ( u ) = υ ( K u ) = 12 Z S n − | u · v | dS K ( v ) , u ∈ S n − . More generally, if K , · · · , K n − ∈ K n , then the mixed projection body of K , · · · , K n − ,Π( K , · · · , K n − ), is an origin-symmetric convex body defined by (see, e.g., [9, 10]) h Π( K , ··· ,K n − ) ( u ) = υ ( K u , · · · , K un − ) = 12 Z S n − | u · v | dS ( K , · · · , K n − ; v ) , u ∈ S n − . Obviously, Π( K, · · · , K ) = Π K .Let ω n = π n/ / Γ(1 + n ) denote the volume of unit ball B in E n for n ≥
0. If K ∈ K no , the L p projection body Π p K ( p ≥
1) of K is an origin-symmetric convex body defined by [14](2.3) h p Π p K ( u ) = 1 nω n c n − ,p Z S n − | u · v | p dS p ( K, v ) , u ∈ S n − , where c n,p = ω n + p /ω ω n ω p − such that Π p B = B .For p ≥ i = 0 , , · · · , n −
1, the L p -mixed projection body of K ∈ K n , Π p,i K , is anorigin-symmetric convex body introduced by Wang and Leng [21].If K ∈ K no , Q ∈ K n , for p ≥ ≤ t ≤ n −
1, the mixed L p projection body is definedby(2.4) h p Π p,t ( K,Q ) ( u ) = 1 nω n c n − ,p Z S n − | u · v | p dS p,t ( K, Q ; v ) , where S p,t ( K, Q ; · ) is the mixed L p -surface area measure of K, Q defined in Section 3. From(2.4), we have Π p,t ( B, B ) = B .If K ∈ K no , the polar body K ∗ of K is defined by K ∗ = { x ∈ E n : x · y ≤ f or all y ∈ K } . A compact set K in E n is star-shaped (about the origin) whose radial function is defined for x = 0 by ρ K ( x ) = max { λ ≥ λx ∈ K } . Obviously, for x = 0 and φ ∈ SL ( n ), ρ φK ( x ) = ρ K ( φ − x ). For K ∈ K n , we can get that(2.5) ρ K ∗ = 1 /h K , and h K ∗ = 1 /ρ K . IXED L p PROJECTION INEQUALITY 5
From (2.5), for K ∈ K n and φ ∈ SL ( n ), we have(2.6) ( φK ) ∗ = φ − t K, where φ − t denotes the inverse of the transpose of φ .If ρ K is positive and continuous, K is called a star body (about the origin). Let S no denotethe set of star bodies (about the origin) in E n . For K, L ∈ S no , p ≥ ε >
0, the L p -harmonic radial combination K + − p ε · L ∈ S no is defined by (see [13]) ρ ( K + − p ε · L, · ) − p = ρ K ( · ) − p + ερ L ( · ) − p . If K, L ∈ S no , for p ≥
1, the L p -dual mixed volume, ˜ V − p ( K, L ), of the K and L is definedby ˜ V − p ( K, L ) = − pn lim ε → o + V ( K + − p ε · L ) − V ( K ) ε . By the polar coordinate formula, one can get the following integral representation of the L p -dual mixed volume ˜ V − p ( K, L ):(2.7) ˜ V − p ( K, L ) = 1 n Z S n − ρ n + pK ( v ) ρ − pL ( v ) dS ( v ) . For φ ∈ SL ( n ), one can get that ˜ V − p ( φK, φL ) = ˜ V − p ( K, L ) or equivalently(2.8) ˜ V − p ( φK, L ) = ˜ V − p ( K, φ − L ) . Obviously, for each star body K ,(2.9) ˜ V − p ( K, K ) = V ( K ) . The dual L p Minkowski mixed volume inequality states that(2.10) ˜ V − p ( K, L ) ≥ V ( K ) ( n + p ) /n V ( L ) − p/n , with equality if and only if K and L are dilates. From the dual L p Minkowski mixed volumeinequality (2.10) and identity (2.9), if K , K ∈ S no ,(2.11) ˜ V − p ( L, K ) = ˜ V − p ( L, K )for all star bodies L which belong to some class that contains both K and K , then K = K .For general references on convex bodies and the L p Brunn-Minkowski theory, we refer toGruber [5] and Schneider [20].3.
Mixed L p surface area measures The Aleksandrov-Fenchel inequality (see, e.g., [10]), in the form most suitable for ourpurposes, states that(3.1) V s ( K, s + t ; Q, n − s − t ) V t ( L, s + t ; Q, n − s − t ) ≤ V s + t ( K, s ; L, t ; Q, n − s − t ) . By repeated applications of the Aleksandrov-Fenchel inequalities and final application of theMinkowski inequality, Lutwak got the following result (see, also [20], p.398).
Lemma 1. If K i ∈ K n , then (3.2) V n ( K , · · · , K n ) ≥ V ( K ) · · · V ( K n ) , with equality if and only if the K i are homothetic. ZHONGWEN TANG AND LIN SI ∗ For Lemma 1, in [1], Alesker, Dar and Milman gave an alternative proof in which ideasfrom mass transportation were used.To establish the Theorem 1, the following result will be needed. The proof is similar tothat of [12].
Theorem 3. If K, L ∈ K no , Q ∈ K n , then for p ≥ and ≤ t ≤ n − , we have lim ε → + V ( K + p ε · L, t + 1;
Q, n − t − − V ( K, t + 1;
Q, n − t − ε = t + 1 np Z S n − h pL ( u ) h K ( u ) − p dS ( K, t ; Q, n − t − u ) . Proof :
Let K ε = K + p ε · L, and define g : [0 , ∞ ) → (0 , ∞ ) by g ( ε ) = V t +1 ( K ε , t + 1; Q, n − t − . Let lim inf ε → + V ( K ε , t ; K ε ; Q, n − t − − V ( K ε , t ; K ; Q, n − t − ε = L inf , and lim sup ε → + V ( K, t ; K ε ; Q, n − t − − V ( K, t ; K ; Q, n − t − ε = L sup . Since K ε ⊃ K , it follows from the monotonicity of the mixed volume that L inf ≥ L sup ≥ , for all ε . We shall make use of the fact that if f , f , · · · ∈ C ( S n − ), with lim i →∞ f i = f ,uniformly on S n − , and ν , ν , · · · are finite measures on S n − such that lim i →∞ ν i = ν , weakly on S n − , then lim i →∞ Z S n − f i ( u ) dν i ( u ) = Z S n − f ( u ) dν ( u ) . From the definition of the L p Minkowski combination, we havelim ε → + h K ε − h K ε = lim ε → + ( h pK + εh pL ) p − h K ε = 1 p h pL h − pK , uniformly on S n − . By the weak continuity of mixed surface area measures S ( K , · · · , K n − , · )(see, Schneider [20], p281) and the fact that lim ε → + K ε = K in K n (see, [3]), we havelim ε → + V ( K ε , t ; K ε ; Q, n − t − − V ( K ε , t ; K ; Q, n − t − ε = lim ε → + n Z S n − h K ε − h K ε dS ( K ε , t ; Q, n − t − u )= 1 np Z S n − h pL ( u ) h − pK ( u ) dS ( K, t ; Q, n − t − u ) . IXED L p PROJECTION INEQUALITY 7
Similarly, lim ε → + V ( K, t ; K ε ; Q, n − t − − V ( K, t ; K ; Q, n − t − ε = lim ε → + n Z S n − h K ε − h K ε dS ( K, t ; Q, n − t − u )= 1 np Z S n − h pL ( u ) h − pK ( u ) dS ( K, t ; Q, n − t − u ) . Then, we have(3.3) L inf = L sup = 1 np Z S n − h pL ( u ) h − pK ( u ) dS ( K, t ; Q, n − t − u ) . From the definitions of L inf , L sup and the Aleksandrov-Fenchel inequality (3.1), we havethatlim inf ε → + V tt +1 ( K ε , t +1; Q, n − t − V t +1 ( K ε , t + 1; Q, n − t − − V t +1 ( K, t + 1;
Q, n − t − ε ≥ L inf , lim sup ε → + V tt +1 ( K, t +1;
Q, n − t − V t +1 ( K ε , t + 1; Q, n − t − − V t +1 ( K, t + 1;
Q, n − t − ε ≤ L sup . By the continuity of V ( K , · · · , K n ) for K i ∈ K n , i = 1 , · · · , n , g is continuous at 0 (see,Schneider [20], P280). And notice that lim ε → + K ε = K in K no . Thus, the above inequalitiescan be rewritten as V tt +1 ( K, t +1;
Q, n − t −
1) lim inf ε → + V t +1 ( K ε , t + 1; Q, n − t − − V t +1 ( K, t + 1;
Q, n − t − ε ≥ L inf , and V tt +1 ( K, t +1;
Q, n − t −
1) lim sup ε → + V t +1 ( K ε , t + 1; Q, n − t − − V t +1 ( K, t + 1;
Q, n − t − ε ≤ L sup . Since L inf = L sup , then the last two inequalities will imply that g is differentiable at 0, and g (0) t g ′ (0) = L inf = L sup . The differentiability of g at 0 implies that g t +1 is differentiable at0, then we have lim ε → + g t +1 ( ε ) − g t +1 (0) ε = ( t + 1) g t (0) lim ε → + g ( ε ) − g (0) ε , or 1 t + 1 lim ε → + V ( K ε , t + 1; Q, n − t − − V ( K, t + 1;
Q, n − t − ε = L inf = L sup . (cid:3) Suppose K ∈ K no and Q ∈ K n , the mixed L p surface area measure of K, Q is defined by(3.4) dS p,t ( K, Q ; · ) = h − pK ( · ) dS ( K, t ; Q, n − t − · ) . The L p,t mixed volume of K, L ∈ K no , Q ∈ K n can also be defined as(3.5) V p,t ( K, L, Q ) = 1 n Z S n − h pL ( u ) dS p,t ( K, Q ; u ) . ZHONGWEN TANG AND LIN SI ∗ When t = n − L p,t mixed volume induces to the L p mixed volume V p,n − ( K, L, Q ) = V p ( K, L ) = 1 n Z S n − h pL ( u ) dS p ( K, u ) . When p = 1 in (3.5), the L p,t mixed volume induces to the mixed volume of K, L, Q ∈ K n V ,t ( K, L, Q ) = V ( K, t ; L ; Q, n − t − . When Q = B in (3.5), the L p,t mixed volume induces to the L p mixed quermassintegrals W p,i ( K, L ), which has the following integral representation, V p,t ( K, L, B ) = W p,i ( K, L ) = 1 n Z S n − h pL ( u ) dS p,i ( K, u ) , where i = n − t − Proof of Theorem 1 :
Case I: t > V p,t ( K, L, Q ) = 1 n Z S n − h pL ( u ) h − pK ( u ) dS ( K, t ; Q, n − t − u ) ≥ V p ( K, t ; L ; Q, n − t − V − p ( K, t + 1;
Q, n − t − ≥ (cid:16) V tn ( K ) V n ( L ) V n − t − n ( Q ) (cid:17) p (cid:16) V t +1 n ( K ) V n − t − n ( Q ) (cid:17) − p = V t +1 − pn ( K ) V pn ( L ) V n − t − n ( Q ) . To obtain the equality conditions, notice that there is equality in H¨older’s inequality pre-cisely when(3.6) V ( K, t ; L ; Q, n − t − h K = V ( K, t + 1;
Q, n − t − h L , almost everywhere, with respect to the measure S ( K, t + 1;
Q, n − t − · ), on S n − . Equalityin inequality (3.2) holds precisely when there exists an x ∈ E n such that(3.7) V ( K, t ; L ; Q, n − t − h K ( u ) = x · u + V ( K, t + 1;
Q, n − t − h L ( u ) , for all u ∈ S n − and there exists a y ∈ E n such that λh Q ( u ) = y · u + h L ( u ) , λ > u ∈ S n − . Since the support of the measure S ( K, t ; Q, n − t − · ) cannot be containedin the great sphere of S n − orthogonal to x . Hence, from (3.6) and (3.7), we have x = 0 and V ( K, t ; L ; Q, n − t − h K = V ( K, t + 1;
Q, n − t − h L everywhere.Case II: t = 0 IXED L p PROJECTION INEQUALITY 9
Notice that V ( K, L ; Q, n −
1) = V ( Q, L ) and V ( K, Q, n −
1) = V ( Q, K ). Thus, wehave V p, ( K, L, Q ) = 1 n Z S n − h pL ( u ) h − pK ( u ) dS Q ( u ) ≥ V p ( Q, L ) V − p ( Q, K ) ≥ (cid:16) V n − n ( Q ) V n ( L ) (cid:17) p (cid:16) V n − n ( Q ) V n ( K ) (cid:17) − p = V − pn ( K ) V pn ( L ) V n − n ( Q ) . The proof of the equality conditions is the same as Case I. (cid:3)
The classical Minkowski’s problem consists in retrieving K from its surface area measure,and it is well-known that it admits a unique solution up to translations. Corresponding to theclassical Minkowski’s problem, for the mixed L p -surface area measure S p,t ( K, Q ; · ), K ∈ K no , Q ∈ K n , we have the following conjecture. Question 1. If µ is an positive Borel measure on S n − , which is not concentrated on a greatsphere of S n − , and ≤ t ≤ n − , then there exist unique K ∈ K no and Q ∈ K n such that S p,t ( K, Q ; · ) = µ ?4. Mixed L p projection inequalities For p ≥ K ∈ S no , the L p -centroid body Γ p K is an origin-symmetric body whosesupport function is given by (see, e.g., [14]) h p Γ p K ( x ) = 1 c n,p V ( K ) Z K | x · y | p dy = 1( n + p ) c n,p V ( K ) Z S n − | x · v | p ρ n + pK ( v ) dS ( v ) , for all x ∈ E n . The normalization is chosen so that, for the unit ball B in E n , we haveΓ p B = B . For the polar of Γ p K , we will write Γ ∗ p K .A consequence of the definition of the L p -centroid body Γ p K , is that for φ ∈ SL ( n ),(4.1) Γ p φK = φ Γ p K. Obviously, from the definition of Γ p K , for λ >
0, we also have(4.2) Γ p λK = λ Γ p K. For the proof of the Theorem 2, the following lemmas are needed.
Lemma 2. If K ∈ K no , L ∈ S no , Q ∈ K n , for p > and < t < n − , then (4.3) V p,t ( K, Γ p L, Q ) = ω n V ( L ) ˜ V − p ( L, Π ∗ p,t ( K, Q )) . Proof : ∗ From the integral representation (3.5), the definition of the L p -centroid body of L , Fubini’stheorem, (2.5) and (2.7), we have V p,t ( K, Γ p L, Q ) = 1 n Z S n − h p Γ p L ( u ) dS p,t ( K, Q, u )= 1 n Z S n − c n,p V ( L ) Z L | u · x | p dxdS p,t ( K, Q, u )= 1 nc n,p V ( L )( n + p ) Z S n − Z S n − | u · v | p ρ n + pL ( v ) dS ( v ) dS p,t ( K, Q, u )= 1 nc n,p V ( L )( n + p ) Z S n − Z S n − | u · v | p dS p,t ( K, Q, u ) ρ n + pL ( v ) dS ( v )= ω n V ( L ) Z S n − ρ n + pL ( v ) h p Π p,t ( K,Q ) ( v ) dS ( v )= ω n V ( L ) Z S n − ρ n + pL ( v ) ρ − p Π ∗ p,t ( K,Q ) ( v ) dS ( v )= ω n V ( L ) ˜ V − p ( L, Π ∗ p,t ( K, Q )) . (cid:3) In [14], Lutwak, Yang and Zhang obtained the following result for the volume of the L p -centroid body. Lemma 3. If K ∈ S no , for p > , then (4.4) V (Γ p K ) ≥ V ( K ) , with equality if and only if K is an ellipsoid centered at the origin. The following Lemma 4 can be obtained by using the similar methods in [16]. A proof isgiven in the Appendix.
Lemma 4. If K, L ∈ K no , Q ∈ K n , for p > , < t < n − and φ ∈ SL ( n ) , then (4.5) V p,t ( φK, L, φQ ) = V p,t ( K, φ − L, Q ) . A property of the operator Π p,t is as follows.
Lemma 5. If K ∈ K no , Q ∈ K n , for p > , < t < n − and φ ∈ SL ( n ) , then (4.6) Π p,t ( φK, φQ ) = φ − t Π p,t ( K, Q ) , where φ − t denotes the inverse transpose of φ . Proof:
IXED L p PROJECTION INEQUALITY 11
From (4.3), (4.5), (4.1) and (2.8), for each L ∈ S no , we have ω n V − p ( L, Π ∗ p,t ( φK, φQ )) V ( L ) = V p,t ( φK, Γ p L, φQ )= V p,t ( K, φ − Γ p L, Q )= V p,t ( K, Γ p φ − L, Q )= ω n V − p ( φ − L, Φ ∗ p,t ( K, Q )) V ( L )= ω n V − p ( L, φ Π ∗ p,t ( K, Q )) V ( L ) . According to (2.11), for all L ∈ S no , ω n V − p ( L, Π ∗ p,t ( φK, φQ )) V ( L ) = ω n V − p ( L, φ Π ∗ p,t ( K, Q )) V ( L )implies that Π ∗ p,t ( φK, φQ ) = φ Π ∗ p,t ( K, Q ) . And from (2.6), we have the desired equality. (cid:3) If K ∈ K no and Q ∈ K , then for p >
1, 0 < t < n − λ , λ >
0, we have(4.7) Π p,t ( λ K, λ Q ) = λ ( t +1 − p ) /p λ ( n − t − /p Π p,t ( K, Q ) . Proof of Theorem 2:
In (4.3), let L = Π ∗ p,t ( K, Q ) and notice that V − p ( K, K ) = K , we can get ω n = V p,t ( K, Γ p Π ∗ p,t ( K, Q ) , Q ) . Combining with (1.5) and (4.4), we have ω n ≥ V t +1 − p/n ( K ) V p/n (Γ p Π ∗ p,t ( K, Q )) V n − t − /n ( Q ) ≥ V t +1 − p/n ( K ) V n − t − /n ( Q ) V p/n (Π ∗ p,t ( K, Q )) . Then the inequality (1.6) is obtained.From the equality conditions of (1.5), the equality holds in the first inequality of the aboveinequalities if and only if K and Γ p Π ∗ p,t ( K, Q ) are dilates and Q is up to translation anddilate. And from the equality condition of (4.4), the equality holds in the second inequalityof the above inequalities if and only if Π ∗ p,t ( K, Q ) is an ellipsoid centered at the origin. Itis easily seen that if Π ∗ p,t ( K, Q ) is an ellipsoid centered at the origin, then Π p,t ( K, Q ) is anellipsoid. Hence, for λ > φ ∈ SL ( n ),(4.8) Π p,t ( K, Q ) = ( λφ ) − B. Let λ = λ ( t +1 − p ) /p λ ( n − t − /p , from (4.6), (4.7) and (4.8), we haveΠ p,t ( λ φ − t K, λ φ − t Q ) = B. Therefore, we have that K = λ − φ t B, Q = λ − φ t B . Then, from (4.8), (2.6), (4.1), (4.2) andthe fact that Γ p B = B , we can obtain Γ p Π ∗ p,t ( K, Q ) = λφ t B .Thus, we conclude that the equality holds in inequality (1.6) if and only if K and Q aredilate ellipsoids centered at the origin. (cid:3) ∗ Appendix
The proof of the Lemma 4 is similar to that of Corollary 1.3 in [16]. For x ∈ E n and x = 0,let h x i = x/ | x | . Definition 2.
Given a measure µ ( u ) on S n − , for p > , φ ∈ GL ( n ) and f ∈ C ( S n − ) ,define the measure µ ( p ) ( φu ) on S n − by Z S n − f ( u ) dµ ( p ) ( φu ) = Z S n − | φ − u | p f ( h φ − u i ) dµ ( u ) . If K , · · · , K n ∈ K n and φ ∈ SL ( n ), then (see, [20], p282)(4.9) Z S n − h φK n ( u ) dS ( φK , · · · , φK n − , u ) = Z S n − h K n ( u ) dS ( K , · · · , K n − , u ) . For each convex body L , it follows from Definition 2, the homogeneity of h L , (2.2) and(4.9) that Z S n − h L ( u ) dS (1) ( K, t ; Q, n − t − φ t u ) = Z S n − | φ − t u | h L ( h φ − t u i ) dS ( K, t ; Q, n − t − u )= Z S n − h L ( φ − t u ) dS ( K, t ; Q, n − t − u )= Z S n − h L ( u ) dS ( φK, t ; φQ, n − t − u ) . From the fact that if two Borel measures on S n − are equal when integrated against supportfunctions of convex bodies then the measures are identical (see, [16]). Thus for K, Q ∈ K n and φ ∈ SL ( n ), we have(4.10) dS ( φK, t ; φQ, n − t − u ) = dS (1) ( K, t ; Q, n − t − φ t u ) . Proposition 1. If K ∈ K no , Q ∈ K n and p > , then for φ ∈ SL ( n ) , (4.11) dS p,t ( φK, φQ, u ) = dS ( p ) p,t ( K, Q, φ t u ) Proof:
For f ∈ C ( S n − ), from (3.4), (2.2), (4.10), Definition 2, the homogeneity of h K , we have Z S n − f ( u ) dS p,t ( φK, φQ, u ) = Z S n − f ( u ) h − pK ( φ t u ) dS ( φK, t ; φQ, n − t − u )= Z S n − f ( u ) h − pK ( φ t u ) dS (1) ( K, t ; Q, n − t − φ t u )= Z S n − | φ − t u | f ( h φ − t u i ) h − pK ( φ t h φ − t u i ) dS ( K, t ; Q, n − t − u )= Z S n − | φ − t u | p f ( h φ − t u i ) h − pK ( u ) dS ( K, t ; Q, n − t − u )= Z S n − | φ − t u | p f ( h φ − t u i ) dS p,t ( K, Q, u )= Z S n − f ( u ) dS ( p ) p,t ( K, Q, φ t u ) . IXED L p PROJECTION INEQUALITY 13 (cid:3)
Proof of Lemma 4:
From (3.5), Proposition 1, Definition 2, the homogeneity of the support function, and (2.2),we have V p,t ( φK, L, φQ ) = 1 n Z S n − h pL ( u ) dS p,t ( φK, φQ, u )= 1 n Z S n − h pL ( u ) dS ( p ) p,t ( K, Q, φ t u )= 1 n Z S n − | φ − u | p h pL ( h φ − t u i ) dS p,t ( K, Q, u )= 1 n Z S n − h pL ( φ − t u ) dS p,t ( K, Q, u )= V p,t ( K, φ − L, Q ) . (cid:3) References [1] S. Alesker, S. Dar, V. Milman,
A remarkable measure preserving diffeomorphism between twoconvex bodies in R n , Geom. Dedicata, 1999, 74:201-212.[2] S. Campi, P. Grochi, The L p -Busemann-Petty centroid inequality , Adv. Math., 2002, 167:128-141.[3] W.J. Firey, p-means of convex bodies , Math. Scand. Math. Soc., 1962, 10:17-24.[4] E. Grinberg, G. Zhang, Convolutions, transforms, and convex bodies , Proc. London Math. Soc.,1999, 78:73-115.[5] P.M. Gruber,
Convex and discrete geometry , volume 336 of Grundlehren der mathematischenWissenschaften, Springer Berlin, 2007.[6] A.J. Li, Q.Z. Huang,
The L p Loomis-Whitney inequality , Adv. Appl. Math., 2016, 75:94-115.[7] A.J. Li, Q.Z. Huang, D.M. Xi,
Sections and Projections of L p -Zonoids and Their Polars , J. Geom.Anal., 2018, 28:427-447.[8] E. Lutwak, Dual mixed volumes , Pacific J. Math., 1975, 58:531-538.[9] E. Lutwak,
Mixed projection inequalities , Trans. Amer. Math. Soc., 1985, 287:91-106.[10] E. Lutwak,
Volume of mixed bodies , Trans. Amer. Math. Soc., 1986, 294:487-500.[11] E. Lutwak,
Intersection bodies and dual mixed volumes , Adv. Math., 1988, 71:232-261.[12] E. Lutwak,
The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem , J.Differential Geom., 1993, 38:131-150.[13] E. Lutwak,
The Brunn-Minkowski-Firey theory.II.Affine and geominimal surface areas , Adv.Math., 1996, 118:244-294.[14] E. Lutwak, D. Yang, G. Zhang, L p affine isoperimetric inequalities , J. Differ. Geom., 2000, 56:111-132.[15] E. Lutwak, D. Yang, G. Zhang, Volume inequalities for subspaces of L p , J. Differ. Geom., 2004,68:159-184.[16] E. Lutwak, D. Yang, G. Zhang, L p John ellipsoids , Proc. Lond. Math. Soc., 2005, 90:497-520.[17] E. Lutwak, D. Yang, G. Zhang, L p dual curvature measures , Adv. Math., 2018, 329:85-132.[18] S.J. Lv, L ∞ Loomis-Whitney inequalities , Geom. Dedicata, 2019, 199:335-353.[19] C.M. Petty,
Isoperimetric problems , Proc. Conf. Convexity and Combinatorial Geometry (Univ.Oklahoma, 1971), University of Oklahoma, 1972, pp. 26-41.[20] R. Schneider,
Convex bodies: the Brunn-Minkowski theory , Encyclopedia of Mathematics and ItsApplications, Vol.151, Cambridge University Press, Cambridge, 2014. ∗ [21] W.D. Wang, G.S. Leng, The Petty projection inequality for L p -mixed projection bodies , Acta MathSinica (English Series), 2007, 23:1485-1494.[22] G. Zhang, Dual kinematic formulas , Trans. Amer. Math. Soc., 1991, 351:985-995.[23] G. Zhang,
Centered bodies and dual mixed volumes , Trans. Amer. Math. Soc., 1994, 345:777-801.[24] D. Zou, G. Xiong,
Orlicz-John ellipsoids , Adv. Math., 2014, 265:132-168.
College of Science, Beijing Forestry University, Beijing 100083, P.R.China
E-mail address : [email protected] College of Science, Beijing Forestry University, Beijing 100083, P.R.China
E-mail address ::