aa r X i v : . [ m a t h . M G ] S e p The mixed affine quermassintegrals ∗ Chang-Jian Zhao
Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. ChinaEmail: [email protected]
Abstract
In this paper, we introduce first the mixed affine quermassinte-grals of j convex bodies. The Aleksandrov-Fenchel inequality for the mixedaffine quermassintegrals of j convex bodies is established. As a application,the Minkowski’s, Brunn’s Minkowski’s inequalities for the mixed affine quer-massintegrals are also derived. Keywords convex body, affine quermassintegrals, Minkowski inequality,Aleksandrov-Fenchel inequality.
1. Introduction
Lutwak [1] proposed to define the affine quermassintegrals for a convex body K , Φ ( K ), Φ ( K ), . . . , Φ n ( K ), by taking Φ ( K ) := V ( K ) , Φ n ( K ) := ω n and for 0 < j < n ,Φ n − j ( K ) := ω n "Z G n,j (cid:18) vol j ( K | ξ ) ω j (cid:19) − n dµ j ( ξ ) − /n , (1 . G n,j denotes the Grassman manifold of j -dimensional subspaces in R n , and µ j denotes the gaugeHaar measure on G n,j , and vol j ( K | ξ ) denotes the j -dimensional volume of the positive projection of K on j -dimensional subspace ξ ⊂ R n and ω j denotes the volume of j -dimensional unit ball (see [2]). Lutwakshowed the Brunn-Minkowski inequality for the affine quermassintegrals. If K and L are convex bodiesand 0 < j < n , then Φ n − j ( K + L ) /j ≥ Φ n − j ( K ) /j + Φ n − j ( L ) /j . (1 . j convex bodies. The Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals of j convex bodies is established. As a applica-tion, and the Minkowski inequality is also derived. ∗ Research is supported by National Natural Sciences Foundation of China (10971205, 11371334). . The mixed affine quermassintegrals In the section, we introduce first the following concept and List its properties and related inequalities.
Definition 2.1 (The mixed affine quermassintegrals of j convex bodies) The mixed affine quermass-integral of j convex bodies K , . . . , K j , denoted by Φ n − j ( K , . . . , K j ), defined byΦ n − j ( K , . . . , K j ) := ω n "Z G n,j (cid:18) vol j (( K , . . . , K j ) | ξ ) ω j (cid:19) − n dµ j ( ξ ) − /n , (2 . ≤ j ≤ n .When K = · · · = K j = K , Φ n − j ( K , . . . , K j ) becomes Lutwak’s affine quermassintegral Φ n − j ( K ).When K = · · · = K j − = K and K j = L , Φ n − j ( K , . . . , K j ) becomes a new affine geometric quantity,denoted by Φ n − j ( K, L ) and call it mixed affine quermassintegral of K and L . When K = · · · = K j − i − = K , K j − i +1 = · · · = K j − = B and K j − i = L , Φ n − j ( K , . . . , K j ) becomes another new affine geometricquantity, denoted by Φ n − j,i ( K, L ) and call it i -th mixed affine quermassintegral of K and L , where0 ≤ i < j ≤ n . When K = · · · = K j − i = K and K j − i +1 = · · · = K j = B , Φ n − j ( K , . . . , K j ) becomes anew affine geometric quantity, denoted by Φ n − j,i ( K ) and call it i -th mixed affine quermassintegral of K ,where 0 ≤ i < j ≤ n .Obviously, the mixed affine quermassintegrals of j convex bodies is invariant under simultaneousunimodular centro-affine transformation. Lemma 2.1 If K , . . . , K j ∈ K no and ≤ j ≤ n , then for any g ∈ SL(n)Φ n − j ( gK , . . . , gK j ) = Φ n − j ( K , . . . , K j ) . As we all know, according to the Brunn-Minkowski theory, a very natural question is raised: arethere some isoperimetric inequalities about the mixed affine quermassintegrals of j convex bodies? Thefollowing perfectly answers the question and establish Minkowski’s, and leksandrov-Fenchel’s and Brunn-Minkowski’s inequalities for the mixed affine quermassintegrals. Theorem 2.1 (The Minkowski inequality for mixed affine quermassintegrals) If K, L ∈ K no and ≤ j ≤ n , then Φ n − j ( K, L ) j ≥ Φ n − j ( K ) j − Φ n − j ( L ) , (2 . with equality if and only if K and L are homothetic.Proof This follows immediately from the Minkoweski’s, and H¨older’s inequalities. (cid:3)
Next, we establish an Aleksandrov-Fenchel inequality for the mixed affine quermassintegral of j convexbodies K , · · · , K j . 2 heorem 2.3 (The Aleksandrov-Fenchel inequality for mixed affine quermassintegrals of j convexbodies) If K , · · · , K j ∈ K no , ≤ j ≤ n and < r ≤ j , then Φ n − j ( K , · · · , K j ) ≥ r Y i =1 Φ n − j ( K i , · · · , K i , K r +1 , · · · , K j ) /r . (2 . Proof
This follows immediately from the Aleksandrov-Fenchel inequality and H¨older’s inequality. (cid:3)
Unfortunately, the equality conditions of the Aleksandrov-Fenchel inequality are, in general, unknown.
Corollary 2.1 If K , · · · , K j ∈ K no and ≤ j ≤ n , then Φ n − j ( K , · · · , K j ) j ≥ Φ n − j ( K ) · · · Φ n − j ( K j ) , (2 . with equality if and only if K , · · · , K j are homothetic. Proof
The special case r = j −
1, of inequality (4.3), isΦ n − j ( K , · · · , K j ) j − ≥ Φ n − j ( K , K j ) · · · Φ n − j ( K j − , K j ) . When above inequality is combined with the Minkowski inequality (4.2), the result isΦ n − j ( K , · · · , K j ) j ≥ Φ n − j ( K ) · · · Φ n − j ( K j ) , with equality if and only if K , · · · , K j are homothetic. (cid:3) Finally, we simply prove the Brunn-Minkowski inequality for the affine quermassintegrals by usingthe mixed affine quermassintegrals theory introduced in this section.
Theorem 2.3 (The Brunn-Minkowski inequality for affine quermassintegrals) If K, L ∈ K no and ≤ j ≤ n , then for ε > n − j ( K + ε · L ) /j ≥ Φ n − j ( K ) /j + ε Φ n − j ( L ) /j . (2 . Proof
This follows immediately from (2.1) and (2.2). (cid:3)
References [1] E. Lutwak, A general isepiphanic inequality,
Proc. Amer. Math. Soc. , (1984), 451-421.[2] E. Lutwak, Inequalities for Hadwigers harmonic quermassintegrals, Math. Ann. , (1988), 165-175.(1988), 165-175.