LET K n denote the set of convex bodies (compact,

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1 The Geeral L -Dual Mixed Brightess Itegrals Pig Zhag, Xiaohua Zhag, ad Weidog Wag Abstract Based o geeral L -mixed brightess itegrals of covex bodies ad geeral L -itersectio bodies of star bodies, this aer is goig to defie the geeral L -dual mixed brightess itegrals. After studyig their extremum values ad establishig Aleksadrov-Frechel iequality, cyclic iequality ad the Bru-Mikowski iequality for the geeral L -dual mixed brightess itegrals, we obtai a more geeral result tha the Bru-Mikowski iequality for the geeral L -dual mixed brightess itegrals. Idex Terms geeral L -mixed brightess itegrals, geeral L -itersectio body, geeral L -dual mixed brightess itegrals. I. INTRODUCTION LET K deote the set of covex bodies (comact, covex subsets with oemty iteriors) i Euclidea sace R. For the set of covex bodies cotaiig the origi i their iteriors i R, we write Ko. So deotes the set of star bodies (about the origi) i R. Let S deote the uite shere i R, ad let V (K) deote the -dimesioal volume of body K. For the stadard uit ball B i R, we write ω V (B). If K K, the its suort fuctio, h K h(k,.) : R (, ), is defied by (see [2]) h(k, x) max{x y : y K}, x R, where x y deotes the stadard ier roduct of x ad y. For a comact set K i R, which is star shaed with resect to the origi, the radial fuctio, ρ K (u) ρ(k, u), of K is defied by (see [2]) ρ K (u) max{λ 0 : λu K}, u S. () If ρ K is ositive ad cotiuous, the K will be called a star body (about the origi). Two star bodies K ad L are said to be dilates (of oe aother) if ρ K (u)/ρ L (u) is ideedet of u S. If c > 0 ad K S o, the ρ(ck, ) cρ(k, ). Let GL() deote the grou of geeral (osigular) liear trasformatios, if ϕ GL(), from (), we easily have ρ(ϕk, x) ρ(k, ϕ x), x R \ {0}, (2) where ϕ deotes the reverse of trasformatio ϕ. Mauscrit received July 5, 206; revised November 23, 206. This work was suorted by the Natioal Natural Sciece Foudatios of Chia(Grat No.37224),ad was suorted by the Academic Maistay Foudatio of Hubei Provice of Chia (Grat No.B206030). P. Zhag is with the Deartmet of Mathematics, Chia Three Gorges U- iversity, Yichag, , P. R. Chia (zhagig9978@26.com). X. H. Zhag is with the Deartmet of Mathematics, Chia Three Gorges Uiversity, Yichag, , P. R. Chia (zhagxiaohua07@63.com). W. D. Wag is with the Deartmet of Mathematics, Chia Three Gorges Uiversity, Yichag, , P. R. Chia (wagwd722@63.com). The otio of mixed brightess-itegrals of covex bodies was defied by Li(see [8]). After that, Ya ad Wag exteded mixed brightess-itegrals to the geeral mixed brightess-itegrals of covex bodies: For K,, K Ko, ad [, ], the geeral L -mixed brightess itegrals, D () (K,, K ), of K,, K is defied by (see [22]) D () (K,, K ) (K, u) () (K, u)ds(u), () S where () (K, u) 2 h(π K, u) deotes the half geeral L -brightess of K Ko ad Π K deotes the geeral L - rojectio body of K Ko. Further, they established some iequalities for the geeral L -mixed brightess itegrals(see [22]). Recetly, Wag ad Li used the fuctio φ : R [0, ) which is give by φ (t) t t, [, ] (3) to defie the geeral L -itersectio body with arameter as follows: For K So, 0 < <, ad [, ], the geeral L -itersectio body, I K So, of K is defied by (see [20]) K, u) i() φ (u x) dx, u S, (4) where i() K ( ) ( ) ( ) ( ). I this aer, based o the geeral L -itersectio bodies ad the geeral L -mixed brightess itegrals, we are goig to defie the geeral L -dual mixed brightess itegrals of star bodies as follows: For K,, K S o, 0 < < ad [, ], the geeral L -dual mixed brightess itegrals, D (K,, K ), of K,, K is defied by D(K,, K ) (K, u) (K, u)ds(u), (5) S where (K, u) 2 K, u) deotes the half geeral L - dual brightess of K So i directio u S. If 0, we write D(K,, K ) D (K,, K ) ad (K, u) 2 K, u), the D (K,, K ) (K, u) (K, u)ds(u), S we call D (K,, K ) the L -dual mixed brightess itegrals of K,, K S o. (Advace olie ublicatio: 24 May 207)

2 Let K K i K ad K i K L (i 0,,, ), we write D,i(K, L) D (K, K, L, L). If i is ay real, K, L So, 0 < <, [, ], the the geeral L -dual mixed brightess itegrals, D,i (K, L), of K ad L is defied by D,i(K, L) (K, u) i (L, u) i ds(u). (6) S Let 0 i (6), we write D,i (K, L) D,i(K, L). Let i 0 i (6), we write D,0(K, K) D(K) (K, u) ds(u), (7) S which is called the geeral L -dual brightess itegrals of K. Let 0 i (7), we write D (K) D (K); for ±, we write D (K) D ± (K). I this aer, we will establish the followig iequalities for the geeral L -dual mixed brightess itegrals. Iitially, we give the extremal values of the geeral L - dual mixed brightess itegrals. Theorem.. If K S o, 0 < <, [, ], the D (K) D (K) D ± (K), (8) if K is ot origi-symmetric, there is equality i the left iequality if ad oly if 0 ad equality i the right iequality if ad oly if ±. Furthermore, we establish the followig Fechel- Aleksadrov tye iequality for the geeral L -dual mixed brightess itegrals. Theorem.2. If K,, K S o, 0 < <, [, ] ad < m, the D (K,, K ) m D(K,, K m, K i,, K i ) (9) equality holds if ad oly if I K m,, I K are dilates of each other. Let 0 i Theorem.2, we obtai the followig iequality. Corollary.. If K,, K So, 0 < < ad < m, the D (K,, K ) m D (K,, K m, K i,, K i ), equality holds if ad oly if I K m,, I K are dilates of each other. Additioally, we establish the followig cyclic iequality for the geeral L -dual mixed brightess itegrals. Theorem.3. >, the Iet K, L S o, 0 < <, [, ], if D,i(K, L) D,k(K, L) D,j(K, L), (0) equality holds if ad oly if I K ad I L are dilates. If 0 < <, the iequality (0) is reversed. Let 0 i Theorem.3, we obtai the followig iequality. Corollary.2. Iet K, L So, 0 < <, if >, the D,i (K, L) D,k (K, L) D,j (K, L), () equality holds if ad oly if I K ad I L are dilates. If 0 < <, the iequality () is reversed. Fially, we obtai the Bru-Mikowski tye iequality for the geeral L -dual mixed brightess itegrals as follows: Theorem.4. Iet K, K, L So, 0 < <, [, ], if i, the D,i(K K, L) i D,i(K, L) i D,i (K, L) i, (2) equality holds if ad oly if I K ad I K are dilates. If i, the iequality (2) is reversed. Actually, we rove a more geeral result tha Theorem.4 i Sectio III. Our work belogs to a ew ad raidly evolvig asymmetric L dual Bru-Mikowski theory that has its ow origi i the work of Ludwig, Haberl ad Schuster (see [3], [4], [5], [6], [0], []). For the further researches of asymmetric L Bru-Mikowski theory, we ca refer to aers [], [7], [4], [5], [6], [7], [8], [9], [20], [2]. A. Dual mixed volumes II. PRELIMINARIES I 975, Lutwak (see [9]) gave the otio of dual mixed volumes as follows: For K, K 2,, K So, the dual mixed volume, Ṽ (K, K 2,, K ), of K, K 2,, K is defied by Ṽ (K,, K ) ρ(k, u) ρ(k, u)ds(u). (3) S If K K i K, K i K L i (3), we write Ṽi(K, L) Ṽ (K, i; L, i), where K aears i times ad L aears i times. Let i be ay real, we have Ṽ i (K, L) ρ(k, u) i ρ(l, u) i ds(u). (4) S Let i 0 i (4), the Ṽ 0 (K, L) V (K) ρ(k, u) ds(u). (5) S B. Some L -combiatios For K, L S o, > 0 ad λ, µ 0(ot both zero), the L -radial liear combiatio, λ K µ L S o, of K ad L is defied by(see [2]) ρ(λ K µ L, ) λρ(k, ) µρ(l, ). (6) For ϕ GL(), K, L S o, > 0 ad λ, µ 0 (ot both zero), from (2) ad (6), we have ϕ(λ K µ L, ) λ ϕk µ ϕl. (7) For K, L S o, 0 < <, [, ], from (4),(6) ad a trasformatio to olar coordiate, we obtai (K L), ) K, ) L, ), (8) (Advace olie ublicatio: 24 May 207)

3 I (K L) I K I L. (9) From (7) ad (9), we get I (ϕ(k L)) I ϕk I ϕl. (20) III. PROOFS OF THEOREMS I this sectio, firstly we shall rove Theorems.-.3, the we rove a more geeral result tha Theorem.4, a quotiet form of the Bru-Mikowski tye iequality for the geeral L -dual mixed brightess itegrals. I order to rove Theorem., we eed the followig iequality (see [20]). Lemma 3.. If K S o, 0 < <, [, ], the V (I K) V (I K) V (I ± K). (2) If K is ot origi-symmetric, there is equality i the left iequality if ad oly if 0 ad equality i the right iequality if ad oly if ±. Proof of Theorem.. If K, L So, from (6), the D,i(K, L) (K, u) i (L, u) i ds(u) S 2 K, u) i L, u) i ds(u) S 2 Ṽi(I K, I L). (22) Let i 0 i (22), ad from (5), we have D,0(K, L) D (K) 2 V (I K). Accordig to (2), we get 2 V (I K) 2 V (I K) 2 V (I± K). D (K) D (K) D ± (K). (23) Accordig to (2), we kow that if K is ot origisymmetric, there is equality i the left iequality if ad oly if 0 ad equality i the right iequality if ad oly if ± i (23). Ad (23) is just the iequality (8). The roof of Theorem.2 requires the followig extesio of the Hölder iequality (see [8] [3]). Lemma 3.2. If f 0, f,, f m are (strictly) ositive cotiuous fuctios defied o S ad λ,, λ m are ositive costats the sum of whose recirocals is uity, the f 0 (u)f (u) f m (u)ds(u) S ( ) λ i f 0 (u)f λi i (u)ds(u), (24) S with equality if ad oly if there exist ositive costats α,, α m such that α f λ (u) α mf λ m m (u) for all u S. Proof of Theorem.2. If K,, K So, 0 < <, [, ], < m, ad let λ i m( i m), ad f 0 (u) (K, u) (K m, u), (f 0 if m ), f i (u) (K i, u), ( i m). So ( Accordig to (24), we have (K, u) (K m, u) (K, u)ds(u) S ( (K, u) S (K m, u) (K i, u) m ds(u) ) m. ) m (K, u) (K m, u) (K, u)ds(u) S (K, u) (K m, u) (K i, u) m ds(u). S D (K,, K ) m D(K,, K m, K i,, K i ). (25) The equality coditio i (25) ca be got from the equality coditio i iequality (24) if ad oly if there exist ositive costats α,, α m such that α (K m, u) m α 2 (K m2, u) m α m (K, u) m for all u S. So equality holds i (25) if ad oly if I K m,, I K are dilates of each other. Ad (25) is just the iequality (9). Proof of Theorem.3. Let K, L So, 0 < <, [, ], if >, accordig to (6), ad the Hölder iequality, we have D,i(K, L) D,k (K, L) ( (K, u) i (L, u) i ds(u) S ( (K, u) k (L, u) k ds(u) S ( [ (K, u) ( i) S ( [ (K, u) ( k) S (K, u) ( i) (K, u) S ) ) ) (L, u) i ] ds(u) ) (L, u) k ] ds(u) ( k) (L, u) k (L, u) i ds(u) (K, u) j (L, u) j ds(u) D,j(K, L). S D,i(K, L) D,k (K, L) D,j (K, L). (26) The equality coditio i (26) ca be got from the equality coditio i the Hölder iequalitiy if ad oly if I K ad (Advace olie ublicatio: 24 May 207)

4 I L are dilates. Similarly, if 0 < <, we ca obtai the reverse form of (26). Ad (26) is just the iequality (0). Now, we give a more geeral result tha Theorem.4 as follows: Theorem 3.. For K, L, K So, ϕ GL(), 0 < <, [, ], if 0 j i, the,i (ϕ(k K ) ), L) D,j (ϕ(k K ), L) ) (,i (ϕk, L) D,i (ϕk ), L), (27) D,j (ϕk, L) D,j (ϕk, L) with equality holds i (27) if ad oly if I ϕk ad I ϕk are dilates. If j 0 < i, the iequality (27) is reversed. The roof of Theorem 3. requires the followig Dresher s iequality(see [23]). Lemma 3.3. Let fuctios f, f 2, g, g 2 0, E is a bouded measurable subset i R. If r 0, the ( E(f f 2 ) ) dx E (g g 2 ) r dx ( E f dx ) (E f 2 dx E gr dx E gr 2 dx ), (28) equality holds if ad oly if f f 2 g g 2. If > 0 > r, the iequality (28) is reversed. Proof of Theorem 3.. For K, K, L So, ϕ GL(), 0 < <, [, ], if 0 j i, accordig to (6),(20) ad (28), we have,i (ϕ(k K ) ), L) D,j (ϕ(k K ), L) ( S (ϕ(k K ), u) i (L, u) i ds(u) S (ϕ(k K ), u) j (L, u) j ds(u) ( S (ϕ(k K )), u) i L, u) i ds(u) S (ϕ(k K )), u) j L, u) j ds(u) ( S ϕk, u) i L, u) i ) ds(u) S ϕk, u) j L, u) j ds(u) ( S ϕk, u) i L, u) i ) ds(u) S ϕk, u) j L, u) j ds(u) ),i (ϕk, L) ( D,i (ϕk ), L). D,j (ϕk, L) D,j (ϕk, L) ) The equality coditio i (27) ca be got from the equality coditio i (28) if ad oly if I ϕk ad I ϕk are dilates. If j 0 < i, similarly, we ca rove that the reverse of the iequality (27) is true. If ϕ is idetic, the we get the followig iequality. Theorem 3.2. For K, L, K So, 0 < <, if 0 j i, the,i (K K, L) D,j (K K, L),i (K, L) D,j (K, L) ) ),i (K, L) D,j (K, L) ), (29) ) with equality holds i (29) if ad oly if I K ad I K are dilates. If j 0 < i, the iequality (29) is reversed. Proof of Theorem.4. Let j i the iequality (29) ad otice that D,(M, L) D(L) by (6), for i ad ay L So, we get D,i(K K, L) i D,i(K, L) i D,i (K, L) i, (30) which is just the iequality (2). From the equality coditio of (29), we see that equality holds i (30) if ad oly if I K ad I K are dilates. Similarly, let j i the reverse of the iequality (29), ad for i, we ca obtai that the reverse of the iequality (30) is true. ACKNOWLEDGMENT The author is idebted to the editors ad the aoymous referees for may valuable suggestios ad commets. REFERENCES [] Y. B. Feg ad W. D. Wag, Geeral L -harmoic Blaschke bodies, Proceedigs of the Idia Academy of Scieces - Mathematical Scieces, vol. 24, o.,. 09-9, Feb [2] R. J. Garder, Geometric Tomograhy, 2d ed., Cambridge Uiversity Press, Cambridge, [3] C. Haberl ad M. Ludwig, A characterizatio of L itersectio bodies, Iteratioal Mathematics Research Notices, Art ID 0548, [4] C. Haberl, L itersectio bodies, Advaces i Mathematics, vol. 27, o. 6, , Ar [5] C. Haberl ad F. E. Schuster, Geeral L affie isoerimetric iequalities, Joural of Differetial Geometry, vol. 83,. -26, Se [6] C. Haberl ad F. E. Schuster, Asymmetric affie L Sobolev iequalities, Joural of Fuctioal Aalysis, vol. 257, o. 3, , Aug [7] C. Haberl ad F. E. Schuster, ad J. Xiao, A asymmetric affie Pólya- Szegö ricile, Mathematische Aale, vol. 352, o. 3, , Mar [8] N. Li ad B. C. Zhu, Mixed brightess-itegrals of covex bodies, Joural of the Korea Mathematical Society, vol. 47, o. 5, , Se [9] E. Lutwak, Dual mixed volumes, Pacific Joural of Mathematics, vol. 58, o. 2, , Ju [0] E. Lutwak, Itersectio bodies ad dual mixed volumes, Advaces i Mathematics, vol. 7, o. 2, , Oct [] M. Ludwig, Mikowski valuatios, Trasactios of The America Mathematical Society, vol. 357, o. 0, , Oct [2] T. Y. Ma ad Y. B. Feg, Some iequalities for -geomiimal surface area ad related results, IAENG Iteratioal Joural of Alied Mathematics, vol. 46, o., , 206. [3] T. Y. Ma ad W. D. Wag, Some iequalities for geeralized L - mixed affie surface areas, IAENG Iteratioal Joural of Alied Mathematics, vol. 45, o. 4, , 205. [4] L. Paraatits, SL()-covariat L -Mikowski valuatios, Joural of the Lodo Mathematical Society, vol. 89, o. 2, , Ja [5] F. E. Schuster ad M. Weberdorfer, Volume iequalities for asymmetric Wulff shaes, Joural of Differetial Geometry, vol. 92, o., , Oct [6] W. D. Wag ad X. Y. Wa, Shehard tye roblems for geeral L -rojectio bodies, Taiwaese Joural of Mathematics, vol. 6,o. 5, , Oct [7] W. D. Wag ad Y. B. Feg, A geeral L -versio of Petty s affie rojectio iequality, Taiwaese Joural of Mathematics, vol. 7, o. 2, , Ar [8] W. D. Wag ad T. Y. Ma, Asymmetric L -differece bodies, Proceedigs of the America Mathematical Society, vol. 42, o. 7, , Mar [9] W. D. Wag ad Y. N. Li, Busema-Petty roblems for geeral L - itersectio bodies, Acta Mathematica Siica, Eglish Series, vol. 3, o. 5, , May (Advace olie ublicatio: 24 May 207)

5 [20] W. D. Wag ad Y. N. Li, Geeral L -itersectio bodies, Taiwaese Joural of Mathematics, vol. 9, o. 4, , Aug [2] M. Weberdorfer, Shadow systems of asymmetric L zootoes, Advaces i Mathematics, vol. 240, , Ju [22] L. Ya ad W. D. Wag, The Geeral L -mixed brightess itegrals, Joural of Iequalities ad Alicatios, vol. 205, o. 90,.-, Ju [23] P. Zhag ad W. D. Wag, ad X. H. Zhag. O dual mixed quermassitegral quotiet, Joural of Iequalities ad Alicatios, vol. 205, o. 340,.-9, Oct (Advace olie ublicatio: 24 May 207)