Rogers-Shephard inequality for log-concave functions

a Xi :1410.2556 2 [ma h.FA] 13 Sep 2016 ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS DAVID ALONSO-GUTI´ ERREZ, BERNARDO GONZ´ ALEZ MERINO, C. HUGO JIM´ ENEZ, RAFAEL VILLA Abs ac . In his pape we p o e di e en unc ional inequali ies ex ending he classical Roge s-Shepha d inequali ies o con ex bodies. The o iginal inequali ies p o ide an op imal ela ion be ween he olume o a con ex body and he olume o se e al symme iza ions o he body, such as, i s di e ence body. We cha ac e ize he equali y cases in all hese inequali ies. Ou me hod is based on he ex ension o he no ion o a con olu ion body o wo con ex se s o any pai o log-conca e unc ions and he s udy o some geome ical p ope ies o hese new se s. 1. In oduc ion A measu e µon Rnis log-conca e i o any measu able se s A, B ⊂Rnand 0< λ < 1, µ(λA + (1 −λ)B)≥µ(A)λµ(B)1−λ whene e A, B ⊂Rnand λA + (1 −λ)Ba e measu able, whe e A+B={a+b: a∈A, b ∈B}is he Minkowski sum. Log-conca e measu es na u ally appea in Con ex Geome y, since he B unn Minkowski inequali y es ablishes he log-conca i y o he Lebesgue measu e e- s ic ed o con ex se s, and o he ma ginal sec ions o con ex se s. A unc ion :Rn→[0,+∞) is log-conca e i (x) = e−u(x) o some con ex unc ion u:Rn→(−∞,∞]. As was shown in [16], a measu e µon Rnwi h ull- dimensional suppo is log-conca e i and only i i has a log-conca e densi y wi h espec o he Lebesgue measu e. The class o log-conca e unc ions has p o en o be o g ea impo ance in se e al a eas o ma hema ics. F om a unc ional poin o iew i has been shown hey esemble Gaussian unc ions in many di e en ways. Many unc ional inequali ies sa is ied by Gaussian unc ions, like Poinca e and Log-Sobole inequali ies, also hold in a mo e gene al subclass o log-conca e unc ions [8, 9, 12]. They also appea in a eas as In o ma ion Theo y, in he s udy o some impo an pa ame e s, such as he classical en opy [14]. The e a e many examples in he li e a u e o unc ional inequali ies wi h a geome ic coun e pa ; P ekopa-Leindle /B unn-Minkowski [27] and Sobole /Pe y p ojec ion [34] inequali ies a e wo o he main examples. This has gene a ed an inc easing in e es o ex end se e al impo an pa ame e s o con ex bodies o unc ional pa ame e s [5, 6, 12, 19, 22, 24] in he class o log- conca e unc ions. Da e: Sep embe 14, 2016. 2010 Ma hema ics Subjec Classi ica ion. P ima y 52A20, Seconda y 39B62,46N10. Key wo ds and ph ases. Roge s-Shepha d inequali y, log-conca e measu es, log-conca e unc- ions, con olu ion body, geome ic inequali ies. 1 2 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA The class o log-conca e unc ions is o en ega ded as he na u al ex ension o con ex bodies, aking in o accoun ha he cha ac e is ic unc ion o a con ex body is a log-conca e unc ion and ha his is he smalles closed unde limi s class o unc ions ha con ains he densi ies o n-dimensional ma ginals o uni o m p obabili ies on con ex bodies o highe dimension (we e e o he nex sec ion o p ecise de ini ions). In his wo k we ex end Roge s-Shepha d inequali y [28, 29] o he class o log- conca e unc ions (Theo ems 2.1 and 2.3), cha ac e izing he equali y cases as well. We p o ide o he unc ional e sions o inequali ies a ound Roge s-Shepha d’s wi h hei espec i e cha ac e iza ion o equali y cases. While he B unn-Minkowski in- equali y is commonly seen as he backbone o mode n Con ex Geome y, Roge s- Shepha d inequali y can be conside ed as a e e se o m o B unn-Minkowski in- equali y ha no only desc ibes a ela ion be ween Minkowski addi ion and olume, bu also deals wi h ye ano he undamen al p ope y in con exi y: symme y. The a eaching in luence o his inequali y becomes e iden as i can be ound as an ing edien no only in many impo an wo ks in classical and asymp o ic con ex geome y [11, 23, 21, 26] bu also in many o he s wi h a mo e analy ical la o [31, 7, 33, 25] and i s ex ension o a unc ional se ing as well as o he en opy o con ex measu es has al eady been conside ed o ins ance in [17] o [15]. The pape is o ganised as ollows: In Sec ion 2 we p o ide he no a ion used in he es o he pape and some p e ious esul s and s a e he p ecise esul s we a e going o p o e. In Sec ion 3 we in oduce he (θ, )-con olu ion bodies o log- conca e unc ions and p o e he ex ension o Roge s-Shepha d inequali ies (8) and (9). In Sec ion 4 we in oduce he mo e gene al concep o k- h (θ, )-con olu ion bodies and p o e a Roge s-Shepha d ype inequali y (10) o su ace a ea. When pa icula izing o k=nwe ob ain he p e iously in oduced (θ, )-con olu ion bodies. In Sec ion 5 we cha ac e ize he equali y cases in hese inequali ies. Finally, in Sec ion 6, we e isi ano he esul a ound Roge s-Shepha d inequali y [17] by gi ing an ex ended e sion o wo di e en unc ions. Thus, we ex end inequali y (7) o any wo log-conca e unc ions and cha ac e ize he equali y cases. Since his inequali y will s eng hen ano he well known Roge s-Shepha d inequali y (4) we will make use o hese new ools o cha ac e ize he equali y cases in inequali y (4). 2. No a ion and p e ious esul s The no a ion used in his pape is qui e s anda d in mode n con ex geome y and consis en wi h o example [32] and [20]. A con ex body is a subse o Rn ha is con ex, compac and has non-emp y in e io . I is said o be cen ally symme ic i o any x∈Kwe ha e ha also −x∈K. When s udying geome ic p ope ies o a con ex body Ki is usually e y con enien o cons uc ano he con ex body om Kwhich is cen ally symme ic. The e a e many ways o cons uc such a symme iza ion. One o hem is he so called di e ence body o K, which is he Minkowski sum o Kand −K. Le us ecall ha he Minkowski sum o wo con ex bodies Kand Lis de ined as K+L={x+y∈Rn:x∈K, y ∈L} ={x∈Rn:K∩(x−L)6=∅}. B unn-Minkowski inequali y (see, o ins ance, [4] o se e al p oo s and cha ac- e iza ion o he equali y cases) gi es he ollowing lowe bound o he olume o ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 3 he Minkowski sum o any wo con ex bodies K, L ⊆Rn: |K+L|1 n≥ |K|1 n+|L|1 n. As a consequence o B unn-Minkowski inequali y he ollowing ela ion be ween he olume o he di e ence body K−Kand he olume o Kis always ue |K−K| ≥ 2n|K|, wi h equali y i and only i Kis symme ic. In [28] Roge s and Shepha d p o ed a e e se inequali y. Namely, Roge s-Shepha d inequali y s a es ha o any con ex body K⊆Rn, we ha e (1) |K−K| ≤ 2n n|K|, wi h equali y i and only i Kis a simplex. This inequali y was ex ended o any pai o con ex bodies K, L ⊆Rnin [29], showing ha (2) max x0∈Rn|K∩(x0−L)||K+L| ≤ 2n n|K||L|, wi h equali y i and only i K=−Lis a simplex (see [2] o he cha ac e iza ion o equali y). In he same pape [29] he au ho s also conside ed di e en ypes o symme iza- ion o a con ex body Kand p o ed olume inequali ies o hem. In pa icula i was shown ha o any con ex body K⊆Rncon aining 0, he olume o he con ex hull o Kand −K e i ies (3) |con {K, −K}| ≤ 2n|K| wi h equali y i and only i Kis a simplex and 0 is one o i s e ices. In he same pape he au ho s ema ked ha , wi h a simila p oo , he la e inequali y can be ex ended o he ollowing inequali y o any wo bodies Kand Lcon aining he o igin (4) |K∩L||con {K, −L}|≤ 2n|K||L| and hey sugges ed ha i is likely ha equali y is a ained i and only i K=Lis a simplex and 0 is one o i s e ices. Ve y ecen ly, in [2], he olume o he θ-con olu ion bodies K+θLwas s udied, whe e (5) K+θL={x∈K+L:|K∩(x−L)| ≥ θmax z∈Rn|K∩(z−L)|}. As a consequence o he olume inequali ies ob ained o con olu ion bodies, in- equali y (2) was eco e ed and he equali y cases we e cha ac e ized. In [1], simila inclusion ela ions and olume inequali ies we e ob ained o he h, θ-con olu ion bodies o Kand L, de ined as K+h,θ L={x∈K+L:h(K∩(x−L)) ≥θmax z∈Rnh(K∩(z−L))}, whe e his a unc ion sa is ying some p ope ies. As a pa icula case we ha e he k- h θ-con olu ion bodies o wo con ex bodies, de ined as K+k,θ L={x∈K+L:Wn−k(K∩(x−L)) ≥θmax z∈RnWn−k(K∩(z−L))}, 4 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA whe e Wn−kdeno es he (n−k)- h que maßin eg al o a con ex body which, by Kubo a’s o mula (c . [32, p. 295]), can be exp essed as an a e age o he olumes o he k-dimensional p ojec ions o K Wn−k(K) = |Bn 2| |Bk 2|ZGn,k |PE(K)|dµ(E). (Gn,k deno es he se o k-dimensional linea subspaces, dµ is he Haa p obabili y measu e on Gn,k and PE(K) is he p ojec ion o Kon a subspace E). As a conse- quence o hese olume inequali ies he ollowing Roge s-Shepha d ype inequali y o any wo con ex bodies K, L ⊆Rn, which in ol es he su ace a ea o Kand L, is ob ained (6) |K+L| ≤ 2n n|K||∂L|+|L||∂K| 2 maxx0∈Rn|∂(K∩(x0−L))|, whe e |∂K|is he su ace a ea o K. No ice ha when L=−Kwe eco e inequal- i y (1). Le us ecall ha , up o a cons an which depends only on he dimension n, he su ace a ea o a con ex body Kequals he que maßin eg al W1(K). Inequali y (3) was ex ended o he con ex o log-conca e unc ions in [17], whe e he au ho p o ed ha o any log-conca e unc ion , i i s di e ence unc ion is de ined by ∆ (z) = sup np (x) (−y) : x, y ∈Rn: 2z=x+yo, hen (7) ZRn ∆ (x)dx ≤2nZRn (x)dx. Taking (x) = e−hK◦(x), wi h hK◦(x) = maxy∈K◦hx, yi he suppo unc ion o he pola se o a con ex body Kcon aining he o igin, inequali y (3) is eco e ed. This inequali y was also ex ended in [3]. In his pape we ex end inequali ies (1), (2) and (6) o log-conca e unc ions. Be o e we s a e ou esul s we need o in oduce some mo e no a ion. Gi en , g wo log-conca e unc ions, hei con olu ion de ined by ∗g(x) = ZRn (z)g(x−z)dz is also a log-conca e unc ion in Rn. I (x) = χK(x) and g(x) = χL(x) a e he cha ac e is ic unc ions o wo con ex bodies, hen ∗g(x) = |K∩(x−L)|. The Asplund p oduc o wo log-conca e unc ions is de ined by ⋆ g(x) = sup z∈Rn (z)g(x−z). I (x) = χK(x) and g(x) = χL(x), hen ⋆ g(x) = χK+L(x). This ope a ion is he na u al ex ension o he Minkowski sum o con ex bodies, as i has been shown when ex ending geome ic inequali ies o he con ex o gene al log-conca e unc ions (see o ins ance [5]). Rema k. No ice ha i bo h and ga e in eg able and con inuous when es ic ed o hei suppo s hen his sup emum is a maximum since, in such case, i ⋆g(x) = 0, hen o any z∈Rnwe ha e ha (z)g(x−z) = 0 and i ⋆ g(x)>0, hen he e exis s a > 0 such ha he se A (x) := {z∈supp ∩(x−supp g) : (z)g(x−z)≥ k k∞kgk∞} ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 5 is no emp y. Since and ga e in eg able log-conca e unc ions his se is con ex and bounded. Thus, i s closu e is a compac con ex se . Since bo h and ga e con inuous when es ic ed o hei suppo s and ⋆ g(x) = sup z∈Rn (z)g(x−z) = sup z∈cl(A (x)) (z)g(x−z) he unc ion (z)g(x−z) is con inuous on he compac se A (x) and he maximum is a ained. Wi h his no a ion, we p o e he ollowing ex ension o inequali y (2). Theo em 2.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo such ha and ga e con inuous when es ic ed o hei suppo s. Then (8) k ∗gk∞ZRn ⋆ g(x)dx ≤2n nk k∞kgk∞ZRn (x)dx ZRn g(x)dx. Fu he mo e, his inequali y becomes an equali y i and only i (x) k k∞=g(−x) kgk∞is he cha ac e is ic unc ion o an n-dimensional simplex. In case we conside g(x) = (−x) he la e inequali y can be imp o ed o he ollowing ex ension o inequali y (1): Theo em 2.2. Le be a log-conca e unc ion wi h ull-dimensional suppo and con inuous when es ic ed o i and le ¯ (x) = (−x). Then (9) ZRn ⋆ ¯ (x)dx ≤2n nk k∞ZRn (x)dx. Fu he mo e, his inequali y becomes an equali y i and only i (x) k k∞is he cha ac- e is ic unc ion o an n-dimensional simplex. Le us men ion ha he p e ious inequali y was i s ob ained by Colesan i [17, Theo em 4.3] whe e he au ho p o es i in he quasi-conca e case wi hou cha ac e izing he equali y case. The ac ha inequali y (9) is an imp o emen o inequali y (8) ollows om Young’s inequali y k ∗¯ k∞≤ k k1k k∞. The no ion o que maßin eg als has also been ex ended om con ex bodies o he se ing o log-conca e unc ions. In [18], [22] and [30], he case o he pe ime e and he mean wid h is conside ed while, in [13], a di e en de ini ion is gi en o all he que maßin eg als. We will wo k wi h he de ini ion in he la e pape , whe e, in pa icula , he que maßin eg al W1(su ace a ea) o a log-conca e unc ion is de ined by W1( ) := Z∞ 0 W1({x∈Rn: (x)≥ })d . By C o on’s o mula (c . [32, p. 235]), his equals W1( ) = cnZAn,1 max z∈E (z)dµn,1(E), whe e cn=|Bn 2| |Bn−1 2|is a cons an depending only on nand An,1is he se o a ine 1-dimensional subspaces o Rnand µn,1is he Haa p obabili y measu e on i . 6 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Theo em 2.3. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo and con inuous when es ic ed o hei suppo s. Then (10) ZRn ⋆ g(x)dx ≤2n nk k∞kgk∞ W1(g)RRn (x)dx +W1( )RRng(x)dx 2 maxx0∈RnW1( (·)g(x0−·)) . Fu he mo e, when n≥3 his inequali y becomes an equali y i and only i (x) k k∞= g(−x) kgk∞is he cha ac e is ic unc ion o an n-dimensional simplex. Finally, we will p o e he ollowing ex ension o (7). Be o e s a ing i le us s a wi h he ollowing de ini ion: De ini ion 2.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions. De ine ⊕g(z) := sup 2z=x+yp (x)g(y) = p ⋆ g(2z). Following he p oo gi en in [17] o inequali y (7), we can show he ollowing esul . The inequali y in he esul was also ob ained in [3] in he mo e gene al case whe e z=λx + (1 −λ)yand no jus λ=1 2. Howe e , equali y cases need a mo e de ailed a gumen . Theo em 2.4. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo s and con inuous when es ic ed o hem. Then (11) ZRnp (x)¯g(x)dx ZRn ⊕g(x)dx ≤2nZRn (x)dx ZRn g(x)dx. Equali y holds i and only i he ollowing wo condi ions a e sa is ied: (i) supp =supp ¯gis a ansla ion o a cone Cwi h e ex a 0 wi h simplicial sec ion, and (ii) (x) = c1e−ha,xion supp and g(x) = c2e−hb,xion supp g o some c1, c2>0 and some a, b ∈Rnsuch ha ha, xi ≥ 0≥ hb, xi o e e y x∈C. 3. (θ, )-con olu ion bodies o log-conca e unc ions and Roge s-Shepha d inequali ies In his sec ion we p o e he a o emen ioned ex ensions o Roge s-Shepha d in- equali y o log-conca e unc ions. In o de o p o e hem we need o in oduce some mo e no a ion. Gi en , g wo in eg able log-conca e unc ions wi h ull- dimensional suppo , x∈supp + supp gand ∈(0,1], le us ecall ha we deno e A (x) = A ( , g)(x) := {z∈supp ∩(x−supp g) : (z)g(x−z)≥ k k∞kgk∞}. Since and ga e in eg able log-conca e unc ions, A (x) is a bounded con ex se . De ini ion 3.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo , ∈(0,1],θ∈[0,1]. We de ine he (θ, )-con olu ion se o and gas he se Cθ, =Cθ, ( , g) := {x∈supp +supp g:A (x)6=∅,|A (x)| ≥ θM } whe e M =M ( , g) := max x0∈supp +supp g|A (x0)|. ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 7 Rema k. When (x) = χK(x) and g(x) = χL(x) a e he cha ac e is ic unc ions o wo con ex bodies Kand L, he se s A (x) = K∩(x−L) o any ∈(0,1] and we eco e he de ini ion o he θ-con olu ion bodies K+θLin [2]. I is ob ious om he de ini ion ha o any ixed , he se s Cθ, dec ease on θ. The ollowing lemma implies he con exi y o hese se s and gi es a e e se (inc easing on θ) inclusion ela ion when no malized by he igh ac o , as shows Co olla y 3.2. Lemma 3.1. Le ∈(0,1], , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo such ha M =|A (0)|,θ1, θ2, λ1, λ2∈[0,1] wi h λ1+λ2≤1. Then λ1Cθ1, +λ2Cθ2, ⊆ Cθ, , wi h 1−θ1 n=λ1(1 −θ 1 n 1) + λ2(1 −θ 1 n 2). P oo . Le x1∈ Cθ1, ,x2∈ Cθ2, . Fo any z0∈ A (0), z1∈ A (x1), z2∈ A (x2), he log-conca i y o and gimplies ((1 −λ1−λ2)z0+λ1z1+λ2z2)g(λ1x1+λ2x2−(1 −λ1−λ2)z0−λ1z1−λ2z2) ≥( (z0)g(−z0))1−λ1−λ2( (z1)g(x1−z1))λ1( (z2)g(x2−z2))λ2≥ k k∞kgk∞. Thus, A (λ1x1+λ2x2)⊇(1 −λ1−λ2)A (0) + λ1A (x1) + λ2A (x2) and, by B unn-Minkowski inequali y |A (λ1x1+λ2x2)|1 n≥ (1 −λ1−λ2)|A (0)|1 n+λ1|A (x1)|1 n+λ2|A (x2)|1 n≥ (1 −λ1−λ2)M 1 n +λ1θ 1 n 1M 1 n +λ2θ 1 n 2M 1 n = (1 −λ1(1 −θ 1 n 1)−λ2(1 −θ 1 n 2))M 1 n . Consequen ly, λ1x1+λ2x2∈ Cθ, . In pa icula , aking θ1=θ2and λ1+λ2= 1 we ob ain ha hese se s Cθ, a e con ex. Besides Co olla y 3.2. Le ∈(0,1], , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-dimensional suppo such ha M =|A (0)|,0≤θ0≤θ < 1. Then Cθ0, 1−θ 1 n 0⊆Cθ, 1−θ1 n . P oo . Taking θ1=θ2=θ0in he p e ious lemma, we ha e ha o any λ1, λ2∈ [0,1] wi h λ1+λ2≤1 (λ1+λ2)Cθ0, =λ1Cθ0, +λ2Cθ0, ⊆ Cθ, , wi h (λ1+λ2)(1 −θ 1 n 0) = 1 −θ1 n. Thus, o any θ0≤θ≤1, aking λ1+λ2=1−θ1 n 1−θ 1 n 0 we ob ain he esul .  In a simila way we can p o e he ollowing: 8 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Lemma 3.3. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull- dimensional suppo . Then o any 1, 2∈(0,1] and any λ∈[0,1] we ha e M 1 n λ 1 1−λ 2≥λM 1 n 1+ (1 −λ)M 1 n 2. Consequen ly, M is con inuous on (0,1). P oo . Le x1, x2be such ha M i=|A i(xi)| o i= 1,2. Since and ga e log-conca e we ha e ha o any z1∈ A 1(x1), z2∈ A 2(x2) and any λ∈,[0,1] (λz1+ (1 −λ)z2)g(λx1+ (1 −λ)x2−(λz1+ (1 −λ)z2)) ≥( (z1)g(x1−z1))λ( (z2)g(x2−z2))1−λ ≥ λ 1 1−λ 2k k∞kgk∞. Thus, A λ 1 1−λ 2(λx1+ (1 −λ)x2)⊇λA 1(x1) + (1 −λ)A 2(x2). By B unn-Minkowsi inequali y M 1 n λ 1 1−λ 2≥ |A λ 1 1−λ 2(λx1+ (1 −λ)x2)|1 n≥λM 1 n 1+ (1 −λ)M 1 n 2. Consequen ly, he unc ion (s) = M 1 n esis conca e in (−∞,0] and hen i is con in- uous on (−∞,0). Thus, M = n(log ) is con inuous on (0,1).  Le us now p o e inequali y (8). P oo o Theo em 2.1 (inequali y). We can assume, wi hou loss o gene ali y, ha k k∞=kgk∞= 1. By de ini ion o Cθ, , we ha e ha o any ∈(0,1] C0, ={x∈supp + suppg:A (x)6=∅} ={x∈supp + suppg: ⋆ g(x)≥ }. Fo any ∈(0,1], le x0( )∈Rnbe such ha M =|A (x0)|. By Co olla y 3.2 wi h θ0= 0 and g eplaced by g(·+x0), o any θ∈[0,1] (1 −θ1 n)(−x0( ) + C0, )⊆ −x0( ) + Cθ, . Taking olumes and in eg a ing in θ∈[0,1] we ob ain |C0, | ≤ 2n nZ1 0|Cθ, |dθ =2n nZRn |A (x)| M dx. Consequen ly M |C0, | ≤ 2n nZRn|A (x)|dx and, in eg a ing in ∈(0,1] Z1 0 M |C0, |d ≤2n nZ1 0ZRn|A (x)|dxd . The in eg al on he igh -hand side is Z1 0ZRn|A (x)|dxd =ZRnZRn (z)g(x−z)dzdx =ZRn (x)dx ZRn g(x)dx. ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 9 On he o he hand, he in eg al on he le -hand side is Z1 0 M |C0, |d =ZRnZ ⋆g(x) 0 max x0∈Rn|A (x0)|d dx ≥max x0∈RnZRnZ ⋆g(x) 0|A (x0)|d dx = max x0∈RnZRnZRn min { ⋆ g(x), (z)g(x0−z)}dzdx. Since k k∞=kgk∞= 1, bo h quan i ies in he minimum a e smalle han o equal o 1, he minimum is bounded om below by he p oduc and so, his quan i y is bounded om below by max x0∈Rn ∗g(x0)ZRn ⋆ g(x)dx. Thus k ∗gk∞ZRn ⋆ g(x)dx ≤2n nk k∞kgk∞ZRn (x)dx ZRn g(x)dx.  Le us now conside he case in which g(x) = (−x). We will deno e his unc ion ¯ and A ( )(x) := A ( , ¯ )(x). No ice ha o any ∈(0,1) A ( )(0) = z∈Rn: (z)2≥ k k2 ∞ =nz∈Rn: (z)≥√ k k∞o. Analogously, le us deno e M ( ) := M ( , ¯ ) and Cθ, ( ) := Cθ, ( , ¯ ). The ollowing lemma shows ha he maximum alue o |A ( )(x)|is a ained a x= 0. Lemma 3.4. Le :Rn→Rbe an in eg able log-conca e unc ion wi h ull- dimensional suppo , hen o any ∈(0,1), A ( )(x)⊆1 2x+A ( )(0). Consequen ly, M ( ) = |A ( )(0)|. P oo . Since is log-conca e, o any x∈supp −supp A ( )(x) = (z∈Rn:s (z) k k∞ (z−x) k k∞≥√ ) ⊆z∈Rn: (z−1 2x) k k∞≥√  =1 2x+A ( )(0).  The ollowing lemma shows a ela ion be ween he (θ, )-con olu ion bodies o and ¯ and he θ-con olu ion bodies o A ( )(0) and −A ( )(0). 16 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA No ice ha aking θ ending o 1 in (i) we ha e ha o e e y ∈(0,1) he maximum o |A (x)|is only a ained a x0( ). Besides, by he con inui y o M , he con inui y o |A (x0)|in and in x0, (ii) holds i and only i x0( ) is he same o e e y ∈(0,1), and hus we may suppose wi hou loss o gene ali y ha x0( ) = 0. Thus (i) o e e y θ, ∈(0,1), (1 −θ1 n)C0, =Cθ, , (ii) o e e y ∈(0,1), maxx0∈Rn|A (x0)|=|A (0)| (iii) o e e y x, z ∈Rn, min{ ⋆ g(x), (z)g(−z)}= ⋆ g(x) (z)g(−z). Fi s o all no ice ha i g(x) = χK(x) is he cha ac e is ic unc ion o a con ex body, hen A (x) = A 2( )(0) ∩(x−K) and Cθ, =A 2( )(0) +θK, whe e A 2( )(0) +θKdeno es he θ-con olu ion o con ex bodies de ined in (5). Thus, since he (θ, )-con olu ion bodies o he unc ions a e he θ-con olu ion bodies o some con ex bodies we ha e equali y in (i) i and only i o e e y ∈(0,1) A 2( )(0) = −Kis a simplex and, consequen ly (x) = g(−x) is he cha ac e is ic unc ion o a simplex. We will p o e ha in he equali y case necessa ily one o he unc ions is he cha ac e is ic unc ion o a con ex body. Condi ion (iii) occu s i and only i ⋆ g(x) o (z)g(−z) equals 0 o 1, o e e y x, z ∈Rn. Fi s , assume ha ⋆ g(x) = 1 o e e y x∈supp + supp g. Then o e e y x∈supp + supp g,A1( )(0) ∩(x−A1(g)(0)) 6=∅and so A1( )(0) + A1(g)(0) = supp + supp g. Consequen ly and ga e cha ac e is ic unc ions. Le us now assume ha he e exis s x∈supp + supp gsuch ha ⋆ g(x)<1. Then o e e y z∈Rn, (z)g(−z) equals 0,1 and hen, o e e y ∈(0,1], A (0) = A1(0). In such case he unc ion M is cons an in (0,1]. In pa icula i is also con inuous on = 1 and hen (i) also holds o = 1. No ice ha i |A1(0)|=|A (0)|= 0, hen o e e y ∈(0,1) we ha e ha o e e y θ∈(0,1) Cθ, ={x∈supp +suppg :A (x)6=∅}, con adic ing (i). Thus, i we ha e equali y in (8), |A1(0)|>0. Now, i (i) holds, we ix ∈(0,1] and ake x∈supp( ) + supp(g). I x /∈ C0, hen A (x) = ∅. I x∈ C0, hen he e exis s θx∈[0,1] such ha x∈∂Cθx, and x= (1 −θ 1 n x)y o some y∈∂C0, . Thus, we ha e equali y in θ 1 n x|A (0)|1 n=|A (x)|1 n=|A (θ 1 n x0 + (1 −θ 1 n x)y)|1 n ≥θ 1 n x|A (0)|1 n+ (1 −θ 1 n x)|A (y)|1 n≥θ 1 n x|A (0)|1 n. and, by he equali y cases in B unn-Minkowski inequali y, A (x) is homo he ic o A (0). Thus, o e e y x∈Rnand ∈(0,1], A (x) is ei he emp y o a homo he ic copy o A (0). Mo eo e , i we pa icula ize in = 1, we ha e ha A1(x) ={z: (z)g(x−z) = 1}={z: (z) = 1}∩(x+{z:g(−z) = 1}) and A1(0) ={z: (z)g(−z) = 1}={z: (z) = 1}∩{z:g(−z) = 1} ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 17 and o any θ∈[0,1] hen Cθ,1is he θ-con olu ion o he con ex bodies Cθ,1=A1( )(0) +θA1(g)(0). Thus, by P oposi ion 2.10 in [2], he con ex bodies A1( )(0) and −A1(g)(0) a e he same simplex A1(0). Le us now assume ha none o he unc ions , g is a cha ac e is ic unc ion, and we will ind a con adic ion. Since and ga e no cha ac e is ic unc ions he e exis 0 < 1, 2<1 and z1, z2∈Rnsuch ha 1≤ (z1)<1 and 2≤g(−z2)<1. Le us deno e by F1a ace o A1(0), wi h ou e no mal ec o u1and con ained in he hype plane {x∈ Rn:hx, u1i=c1}, such ha A1(0) ⊆ {x∈Rn:hx, u1i ≤ c1}and hz1, u1i> c1. Analogously, le F2be a ace o A1(0), wi h ou e no mal ec o u2and con ained in he hype plane {x∈Rn:hx, u2i=c2}, such ha A1(0) ⊆ {x∈Rn:hx, u2i ≤ c2} and hz1, u2i> c2. Obse e ha he log-conca i y o and gimply ha con {z1,A1(0)} ⊂ A 2 1( )(0) and con {z2,A1(0)} ⊂ A 2 2(¯g)(0). I F1=F2, hen (con {z1,A1(0)}∩con {z2,A1(0)}) A1(0) 6=∅. Le z0be a poin in his in e sec ion. Then z0∈ A 1 2(0) since 1 2≤ (z0)g(−z0)<1. Howe e , his is no possible, since A 1 2(0) = A1(0). I F16=F2, le xbe a ec o wi h a small enough no m and pa allel o he only edge con ained in all n−1 ace s F3,...,Fn+1 o A1(0) di e en om F1and F2 and poin ing om F2 o F1such ha z1/∈x+A1(0) and A1(0) ∩(x+F2)6=∅, (con {z1,A1(0)} {z1,A1(0)})∩(x+F1)6=∅, and A1(0) ∩(x+ con {z2,A1(0)} {z2,A1(0)})6=∅. Obse e ha o any poin zin he i s in e sec ion, i holds 1 2< 1≤ (z)g(x− z), whe eas i z′is on he second, hen 1 2< 2≤ (z′)g(x−z′) and, since xpoin s om F2 o F1,z′does no belong o x+A1(0). Besides, o any o he ace Fi∩(x+Fi)6=∅and o any poin z′′ in Fi∩(x+Fi) we ha e (z′′)g(x−z′′) = 1. These h ee ypes o poin s belong o he se A 0 1(x), which we ha e p o ed ha is a homo he ic copy o he simplex A 1 2(0) = A1(0). Then, since o e e y ace Fi he e exis poin s in (x+Fi)∩A 1 2(x) we ha e ha x+A1(0) ⊆A 1 2(x) and since he poin s z′∈A 1 2(x) (x+A1(0)) he inclusion is s ic . Thus |A 1 2(0)|=|A1(0)|=|x+A1(0)|<|A 1 2(x)|, con adic ing (ii). Thus, o gmus be a cha ac e is ic unc ion and, consequen ly, (x) = g(−x) is he cha ac e is ic unc ion o a simplex.  The p oo o he equali y cases in (10) ollows he same lines. P oo o Theo em 2.3 (Equali y). Wi hou loss o gene ali y we assume ha || ||∞= ||g||∞= 1. 18 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Then, equali y holds in (10) i and only i i holds equali y on each inequali y all along he p oo o (10). Mo e pa icula ly, i x0( ) is such ha M1, =W1(A (x0)) we ha e ha (i) o e e y θ, ∈(0,1), (1 −θ1 n−1)(−x0( ) + Cn−1 0, ) = −x0( ) + Cn−1 θ, , (ii) max x0∈RnZRnZ ⋆g(x) 0 W1(A (x0))d dx =ZRnZ ⋆g(x) 0 max x0∈RnW1(A (x0( )))d dx (iii) o e e y x∈Rn, E ∈An,1, min{ ⋆ g(x),max z∈E (z)g(x0( )−z)}= ⋆ g(x) max z∈E (z)g(x0( )−z) (i ) o e e y θ∈Sn−1and o e e y z, w ∈θ⊥, , s ∈R, max s∈R z(s)gw( −s) = max s∈Rmin{ z(s)kgwk, gw( −s)k zk∞}. Taking θ ending o 1 in (i) we ha e ha o e e y ∈(0,1] he maximum o W1(A (x)) is only a ained a x0( ). Besides, (ii) holds i and only i x0( ) is he same o e e y , and hus we may suppose wi hou loss o gene ali y ha x0( ) = 0. Thus (i) o e e y θ, ∈(0,1), (1 −θ1 n−1)Cn−1 0, =Cn−1 θ, , (ii) o e e y ∈(0,1], maxx0∈RnW1(A (x0)) = W1(A (0)) (iii) o e e y x∈Rn, E ∈An,1, min{ ⋆ g(x),max z∈E (z)g(−z)}= ⋆ g(x) max z∈E (z)g(−z), (i ) o e e y θ∈Sn−1and o e e y z, w ∈θ⊥, , s ∈R, max s∈R z(s)gw( −s) = max s∈Rmin{ z(s)kgwk, gw( −s)k zk∞}. As in he p e ious case, we ha e ha i g(x) = χK(x) is he cha ac e is ic unc ion o a con ex body, hen he (n−1)- h (θ, )-con olu ion bodies o he unc ions a e he (n−1)- h θ-con olu ion bodies o some con ex bodies and, as i was p o ed in [1], i n≥3 we ha e equali y in (i) i and only i o e e y ∈(0,1) A 2( )(0) = −Kis a simplex and, consequen ly (x) = g(−x) is he cha ac e is ic unc ion o a simplex. We will p o e ha in he equali y case necessa ily one o he unc ions is he cha ac e is ic unc ion o a con ex body. Condi ion (iii) occu s i and only i ⋆ g(x) o maxz∈E (z)g(−z) equals 0 o 1, o e e y x∈Rn, E ∈An,1. As we ha e seen in he p e ious case, i ⋆ g(x) = 1 o e e y x∈supp + supp g hen and ga e cha ac e is ic unc ions. Le us now assume ha he e exis s x∈supp + supp gsuch ha ⋆ g(x)<1. Then o e e y E∈An,1maxz∈E (z)g(−z) equals 0 o 1. Consequen ly, o e e y ∈(0,1] A (0) = A1(0) because o he wise he e exis s some ∈(0,1) and some z∈Rnsuch ha ≤ (z)g(−z)<1 and since z /∈ A1(0) he e exis 1-dimensional a ine subspaces passing h ough zand no in e sec ing A1(0) and o all such subspaces we would ha e ≤maxz∈E (z)g(−z)<1. Like be o e, in such case (i) also holds o = 1. No ice ha i W1(A1(0)) = W1(A (0)) = 0, hen o e e y ∈(0,1) we ha e ha o e e y θ∈(0,1) Cθ, ={x∈supp +suppg :A (x)6=∅}, ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 19 con adic ing (i). Thus, i we ha e equali y in (8), W1(A1(0)) >0. Now, i (i) holds, we ix ∈(0,1] and ake x∈supp( ) + supp(g). I x /∈ C0, hen A (x) = ∅. I x∈ C0, hen he e exis s θx∈[0,1] such ha x∈∂Cθx, and x= (1 −θ 1 n x)y o some y∈∂C0, . Thus, we ha e equali y in θ 1 n xW1(A (0)) 1 n−1=W1(A (x)) 1 n−1=W1(A (θ 1 n x0 + (1 −θ 1 n x)y)) 1 n−1 ≥θ 1 n xW1(A (0)) 1 n−1+ (1 −θ 1 n x)W1(A (y)) 1 n−1 ≥θ 1 n xW1(A (0)) 1 n−1, and, by he equali y cases in B unn-Minkowski inequali y o que maßin eg als, i n≥3 hen A (x) is homo he ic o A (0). Thus, o e e y x∈Rnand ∈(0,1], A (x) is ei he emp y o a homo he ic copy o A (0). Pa icula izing a = 1, we ha e ha o any θ∈[0,1] Cn−1 θ,1is he (n−1)- h θ-con olu ion o he con ex bodies Cn−1 θ,1=A1( )(0) +n−1,θ A1(g)(0). Thus, i (i) holds hen by he cha ac e iza ion o he equali y cases in [1] A1( )(0) and −A1(g)(0) a e he same simplex A1(0). Now, i we assume ha nei he o he unc ions , g is a cha ac e is ic unc ion, wi h he same p oo as be o e we ind a con adic ion. Thus, o gmus be a cha ac e is ic unc ion and, consequen ly, (x) = g(−x) is he cha ac e is ic unc ion o a simplex.  6. Colesan i’s inequali y o wo unc ions In his sec ion we show ha Colesan i’s unc ional e sion o Roge s-Shepha d inequali y (7) can be ex ended o he case in which we conside any pai o unc ions and no necessa ily g(x) = ¯ (x). Le us ecall ha simila esul s we e ob ained in [3]. P oo o Theo em 2.4. Fo any z∈Rnsuch ha ⊕g(z)>0 le xz, yz∈Rnbe such ha ⊕g(z) = p (xz)g(yz) wi h 2z=xz+yz. No ice ha xzand yzexis by Rema k 2, since ⊕g(z) = p ⋆ g(2z). Using he log-conca i y o and g, (12) (x)g(z−x)≥p (xz)g(yz)p (2x−xz)g(xz−2x) o e e y x∈Rn. In eg a ing in x∈Rn ∗g(z)≥p (xz)g(yz)ZRnp (2x−xz)¯g(2x−xz)dx =1 2n ⊕g(z)ZRnp (x)¯g(x)dx. In eg a ing in z∈Rnwe inally ob ain ZRn (x)dx ZRn g(x)dx ≥1 2nZRn ⊕g(z)dz ZRnp (x)¯g(x)dx, as wan ed. Le us now cha ac e ize he equali y cases. I (i) and (ii) a e sa is ied, hen he e exis s p∈Rnsuch ha supp = supp ¯g=p+C 20 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA and o e e y z∈supp + supp gwe ha e ha supp ∩(z−supp g) = (p+C)∩(z+p+C) = p+C∩(z+C) and since Chas a simplicial sec ion, his equals supp ∩(z−supp g) = p′+C o some p′∈Rn. We will show ha p′=p 2+xz 2 o xzsuch ha ⊕g(z) = p (xz)g(2z−xz). No ice ha , as be o e, such xzexis s, because o he con inui y p ope ies o and gon hei suppo s. In such case we would ha e ha o e e y z∈supp + supp g supp ∩(z−supp g) = xz 2+1 2(supp ∩supp ¯g) and hen o e e y z∈supp + supp gand e e y x∈supp ∩(z−supp g) •2x−xz∈supp , and •xz−2x∈supp g. Then, inequali y (12) holds wi h equali y o he unc ions (x) = c1e−ha,xi,x∈ supp and g(x) = c2e−hb,xi,x∈supp g o e e y x, z ∈Rn. In o de o show ha o e e y z∈supp + supp gwe ha e p′=p 2+xz 2, no ice ha 2(supp + supp g) = 2C−2C=C−C= supp + supp g and so, 2z∈supp + supp gand o e e y x∈supp ∩(2z−supp g) p (x)g(2z−x) = √c1c2e−ha, x 2ie−hb,z−x 2i =√c1c2e−hb,zie−ha−b, x 2i and so, ⊕g(z) = √c1c2e−hb,zie−min{ha−b, x 2i:x∈supp ∩(2z−supp g)} =√c1c2e−hb,zie−min{ha−b,¯xi:¯x∈p 2+C∩(z+C)}. Since (C∩z+C) = −p+ supp ∩(z−supp g) = −p+p′+Cwe ha e ha min ¯x∈p 2+(C∩z+C)ha−b, ¯xi= min ¯x∈− p 2+p′+Cha−b, ¯xi and since ha−b, xi ≥ 0 o e e y x∈C, he minimum is a ained when ¯x=xz 2= −p 2+p′. Thus p′=p 2+xz 2. Le us now p o e ha (i) and (ii) a e necessa y condi ions o equali y in (11) o hold. We can assume, wi hou loss o gene ali y, ha (x0) = k k∞=kgk∞= g(y0) = 1 and le us w i e (x) = e−u(x)and g(x) = e− (x) o some con ex unc ions u, . No ice ha , since k k∞=kgk∞= 1, uand ake alues in [0,+∞]. Equali y in (11) happens i and only i o e e y z∈supp +supp gand e e y x∈ supp ∩(z−supp g) we ha e equali y in (12). Thus, o e e y z∈supp +supp g he suppo o bo h unc ions as unc ions o x∈Rnmus be he same and so supp ∩(z−supp g) = xz 2+1 2(supp ∩(−supp g)), whe e xzis such ha ⊕g(z) = p (xz)g(2z−xz). No ice ha in pa icula his implies ha supp ∩(−supp g) is ull-dimensional, since we a e assuming ha supp and supp ga e ull-dimensional and hen he e exis s some z∈supp + ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 21 supp gsuch ha supp ∩(z−supp g) is ull-dimensional and hen (supp ∩ (−supp g)) is ull-dimensional. Besides, o e e y z∈supp + supp gwe ha e equali y in (12) o e e y x∈ supp ∩(z−supp g) i and only i u1 2(2x−xz) + xz 2=1 2(u(2x−xz) + u(xz)) 1 2(xz−2x) + yz 2=1 2( (xz−2x) + (yz)) o e e y x∈supp ∩(z−supp g), whe e xz, yz∈Rna e such ha ⊕g(z) = p (xz)g(yz). In pa icula , o z0=x0+y0 2we ha e ha ⊕g(z0) = p (x0)g(y0) and so u1 2(2x−x0) + x0 2=1 2u(2x−x0) 1 2(x0−2x) + y0 2=1 2 (x0−2x). Consequen ly, o e e y x′=x−x0and y′=x0 2−y0 2−x u(x0+x′) = 1 2u(x0+ 2x′) (y0+y′) = 1 2 (y0+ 2y′). Thus, supp ∩(z0−supp g) = x0 2+1 2(supp ∩(−supp g)) is a closed con ex cone x0+C(wi h Ca cone wi h e ex a 0) and so supp ∩(−supp g) = x0+C and o e e y z∈supp + supp g supp ∩(z−supp g) = xz 2+x0 2+C. Fu he mo e, supp g∩(z0−supp ) = z0−supp ∩(z0−supp g) = z0−x0−C is a closed con ex cone y0+C′. This implies ha C′=−C. Besides, i he e ex o he cone is unique, z0−x0=y0and hen y0=−x0which implies z0= 0. I he e ex is no unique hen, since bo h z0−x0and y0a e e ices o he cone y0−C, also y0+ 2(z0−x0−y0) = y0−2z0=−x0and so y0−C=−x0−Cand −x0+C=y0+C. On he o he hand, o any z∈supp + supp gi x∈supp ∩(z−supp g), hen 2x−xz∈x0+C. I equali y holds in (12) hen uis a ine in any segmen ha connec s x0+Cwi h a poin xzand is a ine in any segmen ha connec s −x0−Cand some yz. Le us now ake z∈x0+ supp g. Then −x0+z∈supp gand, since x0∈supp we ha e ha −x0+z∈(z−supp )∩supp g=z−supp ∩(z−supp g) = z−xz 2−x0 2−Cand so xz 2∈x0 2−Cand xz∈x0−C. Consequen ly xz=x0. O he wise, conside he ay om xz ha passes h ough x0. Since xz∈x0−Cany poin pin his ay such ha he segmen [xz, p] con ains x0 22 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA is con ained in x0+Cand since u≥0 is a ine in he segmen [xz, p] and u(x0) = 0 hen u= 0 o any such p, con adic ing he in eg abili y o . Thus, o any z∈x0+ supp gwe ha e ha xz=x0and yz= 2z−x0. No ice ha hen o e e y z∈x0+ supp gwe ha e ha supp ∩(z−supp g) = xz 2+x0 2+C=x0+C as xz=x0. Consequen ly, supp = supp ∩(−supp g) since o e e y y∈supp , as Cis a ull-dimensional cone, we can ake w∈Csuch ha y∈ −w+ (x0+C). Thus, i we ake z=−w∈ −C=x0−(x0+C)⊂x0+ supp g, we ha e ha y∈z+(x0+C)⊂z−supp gand so y∈supp ∩(z−supp g) = supp ∩(−supp g). Consequen ly supp =x0+C⊆ −supp g. Analogously, ake z∈ −x0+ supp . Since −x0∈supp g, we ha e ha x0+z∈ supp ∩(z−supp g) = xz 2+x0 2+Cand, consequen ly yz 2∈ −x0 2+C. Thus yz∈ −x0+C=y0+C. Consequen ly yz=y0. O he wise, conside he ay om yz ha passes h ough y0. Since yz∈y0+Cany poin pin his ay such ha he segmen [yz, p] con ains y0is con ained in y0−Cand since ≥0 is a ine in he segmen [yz, p] and (x0) = 0 hen = 0 o any such p, con adic ing he in eg abili y o g. Thus, o any z∈ −x0+ supp we ha e ha xz= 2z−y0and yz=y0. No ice ha hen o e e y z∈ −x0+ supp we ha e ha supp ∩(z−supp g) = xz 2+x0 2+C=z−y0 2+x0 2+C=z+x0+C, since x0+C=−y0+C. Consequen ly, −supp g= supp ∩(−supp g) since o e e y y∈ −supp g, as Cis a ull-dimensional cone, we can ake w∈Csuch ha y∈ −w+ (x0+C). Thus, i we ake z=w∈C=−x0+ supp gwe ha e ha y+z∈supp ∩(z−supp g) = z+x0+Cand so y∈x0+C= supp . Consequen ly −supp g⊆supp and so −supp g= supp =x0+C. Now le us see ha is a ine on −x0−C. Le us ake x, y ∈ −x0−Cand conside z=y 2+x0 2∈ −C=x0+ supp g. Then, o his z,yz=yand since x∈ −x0−C, is a ine in he segmen ha connec s xand y. Consequen ly, is a ine on −x0−Cand so g(x) = c2e−hb,xion −x0−C. Thus, Cdoes no con ain any s aigh line lsince o he wise, as is a ine and posi i e, i mus be cons an on he line land so Chas only one e ex and x0=−y0and hb, xi<0 o e e y x∈C {0}. Analogously, uis also a ine on x0+Csince o any x, y ∈x0+C, i we ake z=x 2−x0 2∈C=−x0+ supp we ha e ha o his z,xz=x−x0−y0=xand so, uis a ine on x0+Cand (x) = c1e−ha,xion x0+C. Since k k∞= (x0) we ha e ha ha, xi>0 o e e y x∈C. Finally, conside ing he sec ion o Cby a hype plane, since he in e sec ion o his sec ion wi h any o i s ansla es is homo he ic o i sel , he sec ion mus be a simplex.  Rema k. Theo em 2.4 becomes inequali y (7) i g(x) = ¯ (x) because ⊕g(z) = ∆ (z). Mo eo e , i also eco e s (4) when we pa icula ize (x) = e−hK(x), g(x) = e−hL(−x), whe e hKand hLa e he suppo unc ions o wo con ex bodies Kand L ha con ain he o igin. Then ⊕g(x) = e−hK∩L(x)and p (x)g(x) = e−hK−L 2 (x), ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 23 and since ZRn e−hK(x)dx =n!|K◦|, hen (11) becomes |(K∩L)◦|K−L 2◦≤2n|K◦||L◦|. F om he cha ac e iza ion o he equali y cases in (11), his only con e ges o equal- i y when we conside sequences o se s (K◦ n)nand (−L◦ n)ncon e ging o simplices wi h 0 in one o he e ices and he same ou e no mal ec o s a he ace s ha pass h ough he o igin. Taking in o accoun ha (K∩L)◦= con {K◦, L◦}and K−L 2⊂con {K, −L}, i we change he ole o K◦and L◦by Kand −L o simplici y, hen |K∩L||con {K, −L}| ≤ K◦−L◦ 2◦|con {K, −L}| ≤ 2n|K||L|, showing he asse ion and sligh ly s eng hening (4). In o de o ha e equali y in (4) we mus ha e Kand Lbe simplices wi h 0 in one o he e ices and he same ou e no mal ec o s a he ace s ha pass h ough he o igin and K+L 2= con {K, L}. Thus K=Lis a simplex. O he wise he e exis s a di ec ion θ∈Sn−1such ha hK(θ)< hL(θ) (o hL(θ)< hK(θ)). Thus, hK+L 2(θ)< hL(θ)≤hcon {K,L}(θ) (o hK+L 2(θ)< hK(θ)≤hcon {K,L}(θ)). ACKNOWLEDGEMENTS Pa o his wo k was ca ied ou a he ‘Ins i u o de Ma em´a icas de la Uni e - sidad de Se illa’ (IMUS) whe e he D. Alonso was in i ed and B. Gonz´alez ul illed he p og am ‘Ayudas pa a es ancias co as pos doc o ales en el IMUS’, and hey a e hank ul o he in i a ion and o he good wo king condi ions and en i onmen he e. D. Alonso is pa ially suppo ed by ‘Ins i u Uni e si a i de Ma em`a iques i Apli- cacions de Cas ell´o’, Spanish Minis y o Sciences and Inno a ion (MICINN) p ojec MTM2013-42105-P and BANCAJA p ojec P1-1B2014-35. B. Gonz´alez is pa - ially suppo ed by Spanish Minis y o Economy and Compe i i eness (MINECO) p ojec MTM2012-34037. C. H. Jim´enez and R. Villa a e suppo ed by MINECO p ojec MTM2012-30748 and C. H. Jim´enez is also suppo ed by Capes and IMPA. Re e ences [1] Alonso-Gu i´ e ez D., Gonz´ alez Me ino B., Jim´ enez C.H. Volume inequali ies o he i- h con olu ion bodies. J. Ma h. Anal. 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Di e en ial Geom, 53 (1999), no. 1, pp. 183–202. E-mail add ess:alonsod@uniza .es E-mail add ess:bg.me[email p o ec ed]e E-mail add ess:hugojimenez@ma .puc- io.b E-mail add ess: [email protected] Uni e sidad de Za agoza Technische Uni e si ¨ a M¨ unchen Pon i ´ ıcia Uni e sidade Ca ´ olica do Rio de Janei o Uni e sidad de Se illa