Matrix Inequalities in the Theory of Mixed Quermassintegrals and the L p -Brunn-Minkowski Theory

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1 WSEAS TRANSACTIONS o MATHEMATICS Matrx Iequaltes the Theory of Mxed Quermasstegrals ad the L -Bru-Mows Theory JOHN A. GORDON Deartmet of Mathematcs ad Comuter Scece Cty Uversty of New Yor-Queesborough Commuty College Baysde, New Yor, 364 USA jgordo@qcc.cuy.edu Abstract: The Bru-Mows theory s cetral to covex geometrc aalyss, ad mxed quermasstegrals ad mxed -quermasstegrals lay a very mortat role ths theory. Durg the ast quarter of a cetury both duals ad L extesos of ths theory have bee develoed. It s the am of ths reset wor to cotue the develomet of aalogues, for ostve defte symmetrc matrces, of some of the fudametal otatos, varats, ad equaltes of mxed quermasstegrals, mxed -quermasstegrals ad L Bru-Mows theory. Key Words: Mxed determat, Ordary Quermasstegral, Mxed Quermasstegral, Mxed -Quermasstegral Matrx L -sum, Mows equalty, L -Bru Mows equalty. Itroducto Ths aer establshes mortat matrx equaltes that are aalogous to some fudametal equaltes covex geometry. The two fudametal equaltes are the Mows ad Bru-Mows equaltes. The otos of mxed determats, matrx L - sum, ad symmetrc matrx olyomals for ostve defte symmetrc matrces are quoted ad used to establsh these equaltes. 2 Mxed Determats Defto 2. Mxed Determat [] If A,..., A r are ostve defte symmetrc matrces ad λ,..., λ r are oegatve real umbers, the of fudametal mortace s the fact that the determat of λ A + + λ r A r s a homogeeous olyomal of degree λ,..., λ r gve by Dλ A + + λ r A r λ,..., λ DA,..., A where the sum s tae over all -tules of ostve tegers,..., whose etres do ot exceed r. The coeffcet DA,..., A s the mxed determat of the matrces A,..., A ad s uquely determed by the requremet that t be symmetrc ts argumets. The mxed determat DA, A 2,..., A of matrces A, A 2,..., A ca be regarded as the arthmetc mea of the determats of all ossble matrces that have exactly oe row from the corresodg rows of A, A 2,..., A Praayautaa []. Proertes of Mxed Determats The followg roertes for mxed determats are well ow see for examle Praayautaa []. For matrces A,..., A, B, B ad scalars λ,..., λ : 2. DA,..., A DA π,..., A π where π s a ermutato o {, 2,..., }. 2.2 DA,..., A, B + B DA,..., A, B + DA,..., A, B 2.3 Dλ A,..., λ A λ λ DA,..., A Notato: A, B M s,+, where M s,+ s the sace of, symmetrc, ostve, defte matrces, ad 0, we let DA, ; B, DA,..., A, B,..., B }{{}}{{} coes coes for otatoal smlfcato uroses. We ow state a mortat ad useful theorem for our wor. E-ISSN: Volume 5, 206

2 WSEAS TRANSACTIONS o MATHEMATICS Theorem 2.2 Alesadrov [4, 5] Let A,..., A be real symmetrc matrces where A 2,..., A are ostve defte. The D 2 A, A 2, A 3,..., A DA, A, A 3,..., A DA 2, A 2, A 3,..., A. Equalty holds f ad oly f A λa 2 where λ > 0 s a real umber. The form of ths theorem most sutable for our uroses, states that: D s A, s + t; ΦD t B, s + t; Φ D s+t A, s; B, t; Φ where A, B are ostve defte symmetrc matrces ad Φ s ay s t-tule of ostve defte symmetrc matrces. Equalty holds f ad oly f A λb where λ > 0 a real umber. A very useful equalty ca be obtaed by reeated alcatos of the Alesadrov equalty: Lemma 2.3 For A,..., A M s,+, DA DA D A,..., A. Equalty holds f ad oly f A,, 2,..., are scalar multles of each other; that s, A c j A j, where c j > 0, j. A secal case of ths geeral equalty s the Mows equalty. Theorem 2.4 Mows [] If A ad B M s,+ AD B, wth equalty the D A, B D f ad oly f A cb, c > 0, ad D A, B DA, ; B,. We ow rove the matrx aalog of the Bru- Mows theorem from covex geometry. Theorem 2.5 If A, B M s,+ the D A + B D A+D B, wth equalty f ad oly f A cb, where c s a ozero scalar. Proof. DA + B D A + B, A + B DA + B, ; A + B DA + B, ; A + DA + B, ; B DA + B D A + DA + B D B. Thus we obta D A + B D A + D B. We ow rove a Uqueess Theorem smlar to oe roved by Praayautaa []. Theorem 2.6 Uqueess Theorem Suose A, B, C M s,+ the: m-. D A, C D B, C for all C M s,+ les A B m- 2. D A, B D A, C for all A M s,+ les B C. Proof. The roof of arts ad 2 are very smlar, ad so we oly gve the roof of art. The Mows equalty states that D A, B D ADB for A, B M s,+, wth equalty f ad oly f A cb, c > 0. Sce D A, C D B, C, usg C A, we have DA D A, A D B, A D BD A, wth equalty f ad oly f A cb, c > 0. Sce A s ostve defte, DA > 0 ad D A > 0, ad the the last equalty becomes D A D B ad therefore DA DB, wth equalty f ad oly f A cb, c > 0. Smlarly, we ca show that DB DA, wth equalty f ad oly f A cb, c > 0. We the coclude that DA DB ad A cb, c > 0. Ths s ossble f ad oly f c, ad cosequetly A B. 3 Symmetrc Polyomals Iequalty of Elemetary Defto 3. The th elemetary symmetrc olyomals s x o varables x x,..., x are defed by s x x, s 2 x s x <j x x j, s 3 x < < l <j< x x j x,... x l,..., s x The elemetary symmetrc olyomal fuctos evaluated at λ,..., λ, where λ are the egevalues of A, are related to the characterstc olyomal of a matrx. Precsely, f A t DtI A s the characterstc olyomal of the matrx A, the A t t s λt + s 2 λt 2 ± s λ where s λ s λ,..., λ. Defto 3.2 [4] Let A be a matrx. For the dex set α {,..., }, we deote the rcal submatrx that les the rows ad colums of A dexed by α as A[α, α], or brefly, A[α]. The determat of such a rcal submatrx s called a rcal mor. x E-ISSN: Volume 5, 206

3 WSEAS TRANSACTIONS o MATHEMATICS We deote the sum of the dfferet rcal mors of A M s,+ by E A. Therefore E A : J α α J A[α], where J {,..., } I artcular E A a tra ad E A detas x,..., x evaluated at the egevalues λ,..., λ of A equals E A. Defto 3.3 Oerator Mootoe [7] A realvalued cotuous fucto ft defed o a real terval Ω s sad to be oerator mootoe f A B fa fb for all symmetrc matrces A, B of all szes whose egevalues are cotaed Ω. Defto 3.4 Oerator Covex/Cocave [7] A real-valued cotuous fucto ft defed o a real terval Ω s called oerator covex f for ay 0 < ε <, fεa+ εb εfa+ εfb holds for all symmetrc matrces A, B of all szes wth egevalues Ω. f s called oerator cocave f f s oerator covex. Defto 3.5 Postve Ma [7] A ma Φ : M m M s called ostve f t mas ostve semdefte matrces to ostve semdefte matrces: A 0 ΦA 0. M m ad M are the saces of m m ad matrces resectvely. Proof. Sce D / λa, ; I, λd / A, ; I,. It suffces to rove that D / A + B, ; I, D / A, ; I, + D / B, ; I,. Ths ca be obtaed by alyg the Alesadrov equalty 2.4 to the exaso of DA, ; I, as follows: DA + B, ; I, DA, ; B, ; I, 0 0 D A, ; I, D B, ; I, D / A, ; I, + D / B, ; I, The scalar matrx oerator f : A D / A, ; I, s oerator cocave, ad by Theorem 3.7 s oerator mootoe. It s easy to see that Φ : A A I s a utal ostve lear ma. Here deotes the Hadamard roduct of A ad I. Therefore by Theorem 3.8 we have Ths mles D / A I, ; I, I D / A, ; I, I I D / A, ; I, I. Defto 3.6 Utal Ma [6] A ma Φ : M M s called utal f ΦI m I. D / A I, ; I, D / A, ; I, 3.8a Theorem 3.7 Oerator Mootoe ad Oerator Cocave Fuctos [7] A oegatve cotuous fucto o [0, s oerator mootoe f ad oly f t s oerator cocave. Theorem 3.8 [7] Let Φ be a utal ostve lear ma from M m to M ad f a oerator mootoe fucto o [0,. The for every A 0 M s, fφa ΦfA. Theorem 3.9 The ma A D / A, ; I, from M s,+ to 0, s oerator cocave. Sce, t t s a creasg fucto o 0,, the 3.8a s equvalet to DA I, ; I, DA, ; I, Theorem 3.0 [2] Let A M s,+. The s a,..., a s λ,..., λ 3.8b 3.9a 0, where a ad λ,, 2,...,, are dagoal etres ad egevalues of A, resectvely. Proof. The equalty follows from the fact that A DA, ; I, s varat uder smlarty trasformato, artcularly the dagolzato trasformato A P P. Therefore 3.8b s equvalet to D[a j δ j ], ; I, D, ; I, 3.9b where δ j f j ad zero otherwse. Ths gves the desred result 3.9a. E-ISSN: Volume 5, 206

4 WSEAS TRANSACTIONS o MATHEMATICS 4 Alcatos of Symmetrc Polyomals: The Projecto Oerator We defe the matrx equvalet of the rojecto oerator a maer aalogous to Lutwa. [8]. Defto 4. Projecto Oerator The rojecto oerator s defed through the followg lmt: C A B : lm t 0 E A + tb E A t where A, B M s,+ ad E A s the th symmetrc olyomal s λ,..., λ, λ are egevalues of A, ad c s the rojecto oerator,. Our goal s to obta a formula for C. We ca smlfy the calculato of the lmt by wrtg the E,, terms of the trace fucto aled to the arorate owers of matrces. Frst, recall that detλi A λ λ λ <j + <j< λ λ + λ λ j λ 2 <j<<l λ 4 ± λ λ j λ λ 3 λ λ j λ λ l λ λ + c λ + c 2 λ c where λ,, ad c,, are costats. Equatg the coeffcets of λ of the last two les of the equatos mmedately above, yelds c λ tr A. Fbeer [0] has show that the other c, 2, ca be determed smlarly to obta the followg recursve set of equatos c 2 2 [c tra + tra 2 ], c 3 3 [c 2 tra + c tra 2 + tra 3 ]. c [c tra + c 2 tra c tra + tra ] 4. Cosequetly, E A c, E 2 A c 2,..., E A c, where the E A,,..., are the th elemetary symmetrc olyomals s λ,..., λ, λ are egevalues of A, ad the c,, are gve 4.. Usg the formulas for the E A,, the lmt defto for C ad the Uqueess Theorem, we fd that C A + [A +c A 2 +c 2 A 3 + +c I] Wrtg c 0, ths formula ca be wrtte as C A A[ c A ] + E A A recursve formula for C ca the be wrtte as follows: C A I, C A E AI A C + A, where <. 5 Quermasstegrals of Mxed Projecto Bodes Defto 5. Ordary Quermasstegrals For A M s,+, 0, the the ordary Quermasstegral of A, deoted by W A, s the mxed determat DA, ; I,, wth coes of A ad coes of the detty matrx I. Defto 5.2 Mxed Quermasstegrals [8] The mxed Quermasstegrals W 0 A, B, W A, B,..., W A, B, of A ad B M s,+ are defed by W A, B lm ε 0 + W A + εb W A ε It s easy to see that sce W λa λ W A for all 0, W A, A W A. For A, B M s,+ we have [] DA + B 0 DA, ; B, 5.2 E-ISSN: Volume 5, 206

5 WSEAS TRANSACTIONS o MATHEMATICS We ca also exad W A + εb as follows: W A + εb DA + εb, ; I, ε Cosequetly; W A, B 0 DA, ; B, ; I, DA, ; I, + ε DA, ; B, ; I, W A + ε DA, ; B, ; I, W A + εb W A lm lm ε 0 + ε DA, ; B, ; I, ε DA, ; B, ; I, It clearly follows that for all A M s,+, W A, B W B, sce W A, B DB, I,..., I W }{{} B. We recogze the mxed Quermasstegral W 0 A, B as D A, B. Sce W A, B of defto 5.2 s a drectoal dervatve, we ca rewrte t as W A, B d dε W A + εb 5.3 ε0 W A Let A deote the gradet of the fuctoal W A wth resect to the matrx A []. The the drectoal dervatve Defto 5.2 ad 5.3 ca be wrtte as W A, B d dε W A + εb ε0 A W A B 5.4 The th symmetrc olyomal of egevalues λ,..., λ of A s E A W A ad from the defto of the Projecto Oerator we coclude that C A B W A, B A W A B 5.5 If we exted our defto of the Projecto Oerator ad mxed Quermasstegrals to all matrces M the 5.5 gves: C A W A A 5.6 ad from 4.2 we have the followg recursve formulae for W A A : W A A I, W A A + A W + AI + + W A A W + AI A W + A + W + AI A W + A A A where <. From Defto 5.2, wth B I, we have the followg relatosh betwee W A, 0 : W + A lm W A + εi W A [ ] W A [δîĵ ] Aîĵ Aδîĵ Aîĵ î,ĵ W A î Aîĵ 5.7 But accordg to Praayautaa [, 2, 3], W A E-ISSN: Volume 5, 206

6 WSEAS TRANSACTIONS o MATHEMATICS ca be see from the followg exaso: DA + εi 0 0 ε DA, ; I, ε W A 5.8 Dfferetg 5.8 tmes wth resect to ε ad settg ε 0, we obta d DA + εi! dε DA, ; I, ε0! 5.9 ad cosequetly DA, ; I,!! d DA + εi dε ε 5.0 For ostve defte symmetrc matrx A, there always exsts a orthogoal matrx P such that A P P P P T, where P s the matrx whose colums form a orthoormal egebass of A ad s the dagoal matrx whose dagoal etres are the corresodg egevalues of A. Equato 5.0 the yelds W A DA, ; I,! d! dε DP P + εp IP! d D + εi! dε D, ; I, W λ ε0 ε0 λ j λ j 5. where the sums are tae over all -tule of ostve tegers j,..., j whose etres do ot exceed, wth λ j,, from the set of all ostve egevalues of A. Equato 5. tells us that W s varat uder smlarty trasformato A P P. Therefore from ths varat relato ad 5.7 we have the followg mortat relato betwee W A, 0 : W + A W + W λ j j 5. As a llustrato, for A M 3, we have W 0 A DA λ λ 2 λ 3 W A λ λ 2 λ 3 3 λ j W 2 A 2 W 3 A j j 3 λ λ 2 + λ λ 3 + λ 2 λ 3 j λ j W A 3 λ + λ 2 + λ 3 λ j W 2 A 3 j j λ + λ 2 + λ 3 6 L -Sum ad Scalar Multlcato of Matrces Defto 6. L -Sum of Matrces For matrces A, B M s,+ ad we defe the L -sum of A ad B as: A + B A + B /. The commutatvty of + s obvous. For the assocatvty: A + B + C [A + B + C ] [A + B + C ] [A + B + C ] [A + B + C ] A + B + C. Defto 6.2 L Scalar Multlcato For, B M s,+ ad scalar λ, we defe the L scalar multlcato of λ ad B as λ B λ / B. Cosequetly for scalars α, β ad matrces A, B M s,+ : α A + β B αa + βb /. 7 Mxed -Quermasstegrals We defe the matrx equvalet of mxed - Quermasstegral a maer aalogous to Lutwa [9]. E-ISSN: Volume 5, 206

7 WSEAS TRANSACTIONS o MATHEMATICS Defto 7. Mxed -Quermasstegrals For A, B M s,+,, 0 <, the mxed -Quermasstegrals of A, B, deoted W, A, B, we defe by W W A + ε B W A,A, B lm. Clearly, f, A+ εb ad cosequetly the mxed -Quermasstegral W, A, B W A, B ths artcular case. It s easy to see that W, A, A W A for all : W,A, A W A + ε A W A lm W [A + ε] W A lm [ ] + ε + Oε2 W A W A lm ε 0 + ε W A ad we coclude that W, A, A W A for all. Theorem 7.2 a For all A, B M s,+ ad α, β > 0, W, αa, βb α β W, A, B, ad whe ad β, W, αa, B W, A, B. b For all Q, A, B M s,+, W, Q, A + B W, Q, A + W, Q, B. c For all A, B M s,+, γ > 0, W, A, γ B γw, A, B. Now let ε β ε α, W, αa, βb β α α α β α β W, A, B. W A + ε B W A lm ε 0 + ε W A + ε B W A lm ε 0 + ε Ths shows that the fuctoal W, : M s,+ M s,+ 0, s Mows homogeeous of degree ts frst argumet ad Mows homogeeous of degree ts secod argumet. Trvally, whe, β, α > 0, the W, αa, B W, A, B. For art b tae Q, A, B M s,+, W, Q, A + B W Q + ε A + B W Q lm W Q + ε A + ε B W Q + ε A lm + W Q + ε A W Q lm Wrte Q Q + ε A the frst lmt, W, Q, A + ε B W lm Q + ε B W Q W Q + ε A W Q + lm W, Q, A + W, Q, B. Part c s art a wth α ad γ β. Proof. For α, β > 0 ad A, B M s,+ we have Ths result shows that the mxed - Quermasstegral s lear, wth resect to L -sum ad scalar multlcato, ts secod argumet. W, αa, βb W αa + ε βb W αa Defto 7.3 Jotly Cocave Ma [7] Let P be lm the set of ostve semdefte matrces M. s A ma W αa + ε β α lm B W Ψ : P P P m s called jotly cocave f αa ΨλA + λb, λc + λd α W A + βε α B W lm A λψa, C + λψb, D.. for all A, B, C, D 0 ad 0 < λ <. E-ISSN: Volume 5, 206

8 WSEAS TRANSACTIONS o MATHEMATICS The followg lemma roved by Zha [7] s very useful the roof of Theorem 8.. Lemma 7.4 [7] For 0 < r < the ma A, B A r B r M s,+, W 0 A E A λ λ 2 λ, 0 W A E A λ λ, 0 s jotly cocave A, B 0. 8 Some Useful Iequaltes Theorem 8. [7] For A, B, C, D 0 ad, q > wth / + /q, A B + C D A + C B + q D, where A B : [a j b j ] M. Proof. Ths s just the md-ot jot cocavty case λ /2 of Lemma 7.4 wth r /. Theorem 8.2 For X, Y > 0 that s X, Y M s,+ ad ε [0, ] we have εx + εy εx + εy / : ε X + ε Y Proof. Ths s just Theorem 8. wth A ε / X, B ε /q, C ε / Y ad D ε P /q, where [], that s s the matrx that has all of ts etres equal to. The followg theorem roved by Hor [6] s useful the roof of Corollary 8.4. Theorem 8.3 [6] If A, B M are ostve defte symmetrc, the f A B, the deta detb ad tra trb; ad more geerally, f A B, the λ A λ B for all, 2,..., f the resectve egevalues of A ad B are arraged the same creasg or decreasg order. Corollary 8.4 For ay A, B > 0 M s,+, ε [0, ] W ε A + ε B W εa + εb, Proof. Alyg Theorem 8.3 to the equalty Theorem 8.2 ad usg the fact that for ay matrx A. W 3 A E 3 A λ λ 2 λ 3, 3 W 2 A E 2 A λ λ 2, 2 W A E A λ where E A, s the th symmetrc olyomal of egevalues λ,..., λ of A. Theorem 8.5 For ay A, B > 0 M s,+, ε [0, ] W εa + εb εw A + εb, a Equalty holds f ad oly f A cb wth a real umber c > 0. Proof. It suffces to rove that W A + B W A + W B, b wth equalty f ad oly f A cb, c > 0. The equalty s obtaed by alyg the Alesadrov equalty to the exaso of W A + B as follows: W A + B DA + B, ; I, DA, ; B, ; I, 0 D A, ; I, 0 D B, ; I, D / A, ; I, + D / B, ; I, W / A + W / B 8.5c E-ISSN: Volume 5, 206

9 WSEAS TRANSACTIONS o MATHEMATICS whch s 8.5b. For the equalty art, t ca be easly see that f A cb, c > 0 the W / A + B W / Therefore, we oly eed to rove that W / A + B W / A + W / B. A + W / B mles A cb, c > 0. Suose A cb the the Alesadrov equalty Theorem 2.4 s strct, ad alyg t the rocess of gettg 8.5c yelds W / A + B > W / A + W / B whch meas W / A + B W / A + B mles A cb, c > 0. W / Theorem 8.6 If >, α [0, ], A 0, B 0 > 0 wth W A 0 W B 0 the M s,+ W α A 0 + α B 0, Equalty holds f ad oly f A 0 B a Proof. Alyg 8.4 ad 8.5a wth α ε to A 0, B 0 we obta W α A 0 + α B 0 W αa 0 + αb 0, 0. To see the equalty art of we frst set A 0 B 0, W α A 0 + α B 0 W α A 0 + α A 0 W [αa 0 + αa 0 / ] W [A 0 / ] W A 0. Ths roves that A 0 B 0 mles W α A 0 + α B 0. Now we set W α A 0 + α B 0, from 8.6b, we see that ths mles W αa 0 + αb 0, whch tur, by Theorem 8.5 mles A 0 cb 0, but sce we have W A 0 W B 0 therefore c ad A 0 B 0. Ths roves that W α A 0 + α B 0 mles A 0 B 0. Ths comletes the roof. Theorem 8.7 Bru-Mows Iequalty for L -Sum of Matrces If A, B M s,+, 0,, the W A + B W A + W wth equalty f ad oly f A c, B, c > 0. Proof. We aly Theorem 8.6 wth to obta W or A 0 B 0 α W A W B A, B, W A W A + W B W α A 0 + α B 0 W A W A + W B A + W B W A + W B B. B W B W A + W A W A + W B B {[ W W A + W B A / ]/ } + W A + W B B / / A + B w A + W B W / W / [ A + B / W A + W B W A + W B / that s, W / [A + B / ] W A + B ] / W A + W B. E-ISSN: Volume 5, 206

10 WSEAS TRANSACTIONS o MATHEMATICS The suffcecy of the equalty art ca be see by drectly substtutg A c B, c > 0 ad the ecessty of the equalty art ca be roved by cotradcto as follows: Suose A c B the A 0 B 0 whch tur by Theorem 8.6, mles W α A 0 + α B 0 > or W A + B > W A + W B whch s a cotradcto. Ths comletes the roof. Theorem 8.8 Mows Iequalty for L -Sum of Matrces If A, B M s,+, 0,, the W, A, B W AW B wth equalty f ad oly f A c B, c > 0. Proof. Theorem 8.7 mles W / ε A + ε B W / ε A + W / ε B W / ε / A + W / ε / B ε / W A / + ε / W B / εw / ad sce A + εw / B, W ε A + ε B W A lm ε 0 ε W λ A + λ B W A lm λ λ λ W A λ + λ lm B W A λ λ + ε lm ε 0 where ɛ λ λ W A + ɛ B W A + ε, ɛ gɛfɛ g0f0 lm + ɛ, ɛ 0 ɛ where fɛ W A + ɛ B ad gɛ + ɛ gf 0 g0f 0 + g 0f0 W,A, B + W A. The W, A, B W A + W ε A + ε B W A ε W A + lm ε 0 lm ε 0 [ εw A + εw ε W A + [ ] [W A W A W B B] W B W A]. W A Ths gves the equalty art of Theorem 8.8. The suffcecy of the equalty art ca be see by drectly substtutg A c B, c > 0 usg the fact that W, B, B W B. The ecessty of the equalty art ca be show as follows: W, A, B W AW B for A, B M s,+, 0, >, ad the W, Q, A + B W, Q, A + W, Q, B W, Q, A + B W +W B]. Q[W A We ow set A + B equal to Q ad use the fact that W, Q, Q W Q to obta W A+ B W or W A + B W A+B[W A + W A+W B whch s the equalty art of Theorem 8.7 ad that s f ad oly f A c B, c > 0. Ths roves that A, B W AW B mles A W, cḃ, c > 0. Ths comletes the roof. Furthermore, we ca also show that the equaltes of Theorems 8.7 ad 8.8 are equvalet. Sce we have already show that Theorem 8.7 mles Theorem 8.8, B], E-ISSN: Volume 5, 206

11 t suffces to show that Theorem 8.8 mles Theorem 8.7. Sce W, A, B W AW for A, B M s, 0, >, ad the B, W, Q, A + B W, Q, A + W, Q, B W, Q, A + B W +W Q[W B]. A We ow set A + B equal to Q ad use the fact that W, Q, Q W Q to obta or WSEAS TRANSACTIONS o MATHEMATICS W W A + B W A + B [ W A + W A + B W ] B, A + W whch s the equalty of Theorem 8.7 B The lmtg cases of Theorems 8.7 ad 8.8 for the case where hold due to the Alesadrov equalty Theorem 2.4. The well ow Fudametal Iequalty of Mxed Quermasstegrals see [7] stated below s the lmtg case of Theorem 8.8. Theorem 8.9 Fudametal Iequalty of Mxed Quermasstegrals For A, B M s,+ ad 0 <, W A, B W AW B wth equalty f ad oly f A c B, c > 0. Theorem 8.0 Suose 0 < ad A, B M s,+ are such that W A W B. The a If W A W, A, B, for some >, the A B. b If W A W, B, A, for some such that > > the A B. c If W B W, A, B, for some >, the A B. Proof. a Sce W A W, A, B t follows from Theorem 8.8 that W A W, A, B W AW B wth equalty the rght equalty f ad oly f A cḃ, c 0. Ths strg of equaltes mles that or smly W A W B W A W B. But the hyothess W A W B shows that there s fact equalty both equaltes ad that We coclude that A B. W A W B. b Sce W A W, B, A for some, > >, t follows from Theorem 8.8 that W A W, B, A W BW A wth equalty f ad oly f A cḃ, c 0. Ths last equalty mles that or smly W A W B W A W B. The codto W A W B mles that ad hece A B. W A W B c Ths follows detcally from the roof of arts a ad b. Theorem 8. Suose A, B M s,+. If 0 <, ad > ad f W, A, Q W, B, Q for all Q M s,+, the A B. Proof. Set Q A, ad get W A W, A, A W, B, A. Set Q B, ad get W B W, B, B W, A, B. From arts b ad c of the last theorem we obta A B. Theorem 8.2 Suose A, B M s,+ ad 0 <. If ad W A W, B, A, the A c B, c > 0. E-ISSN: Volume 5, 206

12 WSEAS TRANSACTIONS o MATHEMATICS Proof. From the hyothess ad Theorem 8.8 we have W A W, B, A W BW A wth equalty the rght equalty mlyg that A cb, c > 0. Sce, we wll have W A W, B, A W A so W, B, A W A. Hece, A c B, c > 0. Theorem 8.3 Suose A, B M s,+. If 0 <, ad W, A, Q W, B, Q for all Q M s,+. The W, A, Q W, B, Q for all Q M s,+. Proof. From Theorem 8.8 ad the hyothess ths theorem we wll have W, B, Q W Q W, A, Q, 0 <,, for all Q M s,+. Sce W, A, Q W, B, Q for all Q M s,+, combg these equaltes yeld W, A, Q W, B, Q for all Q M s,+. 9 Acowledgmet The author wshes to tha Professor Poramate Praayautaa for hs may excellet suggestos for mrovg the orgal mauscrt. Refereces: [] Poramate Praayautaa, Elltc Bru- Mows Theory, Ph.D. thess, Polytechc Uversty, BrooLy, NY, Jue 2003 [2] Poramate Praayautaa, A roof of S a,..., a S λ,..., λ, 0 for a Postve Defte Symmetrc Matrx A [a j ], usg Mxed Determats, WSEAS TRANSACTIONS ON MATHEMATICS, Issue, Vol. 6, , Jauary [3] Joh Gordo, Poramate Praayautaa, New Matrx Iequaltes for Frey s Exteso of Mows ad Bru-Mows Iequaltes, WSEAS TRANSACTIONS ON MATHEMATICS, Issue 7, Vol. 5, , July [4] A.D. Alesadrov, Zur Theore der gemschte Voluma vo ovexe Körer, IV. De gemschte Dscrmate ud de gemschte Voluma I Russa. Mat. Sbor N.S , [5] Rolf Scheder, Covex Bodes: The Bru- Mows Theory, Cambrdge Uversty Press, New Yor, 993. [6] Roger A. Hor ad Charles R. Johso, Matrx Aalyss, Cambrdge Uversty Press, New Yor, 985. [7] Xg Zh Zha, Matrx Iequaltes Lecture Notes Mathematcs 790, Srger, NY, 200. [8] Erw Lutwa, O Quermasstegrals of Mxed Projecto Bodes, Geometrae Dedcata 33: 5 58, 990. [9] Erw Lutwa, The Bru-Mows- Frey Theory I: Mxed Volumes ad the Mows roblem, Joural of Dfferetal Geometry 38: 3 50, 993. [0] D.T. Fbeer II, Itroducto to Matrces ad Lear Trasformatos, 2d edto, W.H. Freema ad Comay Sa Frascsco ad Lodo, 960. E-ISSN: Volume 5, 206