Zhangs inequality for log-concave functions B. Gonz alez Merino* - PowerPoint PPT Presentation

Zhang’s inequality for log-concave functions B. Gonz´ alez Merino* (joint with D. Alonso-Guti´ errez and J. Bernu´ es) Jena *Author partially funded by Fundaci´ on S´ eneca, proyect 19901/GERM/15, and by MICINN, project PGC2018-094215-B-I00, Spain. Departamento de An´ alisis Matem´ atico, Universidad de Sevilla Convex, Discrete and Integral Geometry, 19th September 2019.

First definitions • If M ⊂ R n , let | M | be the volume (or n-Lebesgue measure) of M

First definitions • If M ⊂ R n , let | M | be the volume (or n-Lebesgue measure) of M • Let B n 2 be the Euclidean unit ball

First definitions • If M ⊂ R n , let | M | be the volume (or n-Lebesgue measure) of M • Let B n 2 be the Euclidean unit ball ,i.e., � � � x ∈ R n : | x | = B n x 2 1 + · · · + x 2 2 = n ≤ 1 .

First definitions • If M ⊂ R n , let | M | be the volume (or n-Lebesgue measure) of M • Let B n 2 be the Euclidean unit ball ,i.e., � � � x ∈ R n : | x | = B n x 2 1 + · · · + x 2 2 = n ≤ 1 . • For any M ⊂ R n , then χ M is the characteristic of M � 1 if x ∈ M χ M : R n → { 0 , 1 } s. t. χ M ( x ) = 0 otherwise.

Geometric & Analytic ineqs. Theorem 1 (Isoperimetric ineq. 1915) Let M ∈ C 1 and M compact. Then n − 1 1 n | B n n | M | 2 | ≤ S ( M ) . n ”=” iff M = B n 2 .

Geometric & Analytic ineqs. Theorem 1 (Isoperimetric ineq. 1915) Let M ∈ C 1 and M compact. Then n − 1 1 n | B n n | M | 2 | ≤ S ( M ) . n ”=” iff M = B n 2 . Theorem 2 (Sobolev ineq. ’38) Let f : M → R with M ∈ C 1 compact and f ∈ C 1 . Then � n − 1 �� � n 1 n n | B n 2 | R n | f | ≤ R n |∇ f | n − 1 n ”=” iff f = χ B n 2 .

Definitions and Properties • K is a convex body, i.e., a convex compact set of R n . Powered by TCPDF (www.tcpdf.org)

Definitions and Properties • K is a convex body, i.e., a convex compact set of R n . Powered by TCPDF (www.tcpdf.org) • K n set of n-dimensional convex bodies.

Definitions and Properties • If x ∈ R n , then x ⊥ is the hyperplane orthogonal to x .

Definitions and Properties • If x ∈ R n , then x ⊥ is the hyperplane orthogonal to x . Moreover, if M ⊂ R n , then P x ⊥ M is the orthogonal projection of M onto x ⊥ .

Definitions and Properties • If x ∈ R n , then x ⊥ is the hyperplane orthogonal to x . Moreover, if M ⊂ R n , then P x ⊥ M is the orthogonal projection of M onto x ⊥ . • The polar projection body Π ∗ ( K ) of K ∈ K n is the unit ball of the norm � x � Π ∗ ( K ) := | x || P x ⊥ K | .

Definitions and Properties • If x ∈ R n , then x ⊥ is the hyperplane orthogonal to x . Moreover, if M ⊂ R n , then P x ⊥ M is the orthogonal projection of M onto x ⊥ . • The polar projection body Π ∗ ( K ) of K ∈ K n is the unit ball of the norm � x � Π ∗ ( K ) := | x || P x ⊥ K | . • The product | K | n − 1 | Π ∗ ( K ) | is an affine invariant.

Petty projection inequality Theorem 3 (Petty 1971) Let K ∈ K n . Then � n +1 n � n 2 Γ | K | n − 1 | Π ∗ ( K ) | ≤ π 2 � n . � n +2 Γ 2 ”=” iff K is an ellipsoid.

Petty projection inequality Theorem 3 (Petty 1971) Let K ∈ K n . Then � n +1 n � n 2 Γ | K | n − 1 | Π ∗ ( K ) | ≤ π 2 � n . � n +2 Γ 2 ”=” iff K is an ellipsoid. Petty projection ineq. ⇒ Isoperimetric ineq.

Zhang’s inequality Theorem 4 (Zhang 1991) Let K ∈ K n . Then � 2 n � ≤ | K | n − 1 | Π ∗ ( K ) | . n n n ”=” iff K is an n-simplex.

Polar projection body of f For every f ∈ W 1 , 1 = { f ∈ L 1 ( R n ) : ∂ x i ∈ L 1 ( R n ) , i = 1 , . . . , n } ∂ f with compact support, let Π ∗ ( f ), the polar projection body of f , be the unit ball of the norm � � x � Π ∗ ( f ) := R n |�∇ f ( y ) , x �| dy .

Functional affine inequalities Theorem 5 (Zhang 1999) Let f ∈ C 1 with compact support. Then � n +1 1 2 Γ � n ≤ π 1 n − 1 | Π ∗ ( f ) | 2 � f � � . � n +2 n 2Γ 2 ”=” iff f = χ E with E an ellipsoid.

Functional affine inequalities Theorem 5 (Zhang 1999) Let f ∈ C 1 with compact support. Then � n +1 1 2 Γ � n ≤ π 1 n − 1 | Π ∗ ( f ) | 2 � f � � . � n +2 n 2Γ 2 ”=” iff f = χ E with E an ellipsoid. Notice that � n +1 � 1 1 1 n π 2 Γ � �� ≤ n n S n − 1 �∇ u f � − n 2 � f � 1 du . � n +2 n � 2Γ n − 1 2

Log-concave functions • f : R n → [0 , ∞ ) is log-concave if f ( x ) = e − u ( x ) for some u : R n → ( −∞ , ∞ ] convex,

Log-concave functions • f : R n → [0 , ∞ ) is log-concave if f ( x ) = e − u ( x ) for some u : R n → ( −∞ , ∞ ] convex, i.e., if f ((1 − λ ) x + λ y ) ≥ f ( x ) 1 − λ f ( y ) λ for every x , y ∈ R n , λ ∈ [0 , 1].

Log-concave functions • f : R n → [0 , ∞ ) is log-concave if f ( x ) = e − u ( x ) for some u : R n → ( −∞ , ∞ ] convex, i.e., if f ((1 − λ ) x + λ y ) ≥ f ( x ) 1 − λ f ( y ) λ for every x , y ∈ R n , λ ∈ [0 , 1]. • F ( R n ) log-concave integrable functions in R n .

Log-concave functions • f : R n → [0 , ∞ ) is log-concave if f ( x ) = e − u ( x ) for some u : R n → ( −∞ , ∞ ] convex, i.e., if f ((1 − λ ) x + λ y ) ≥ f ( x ) 1 − λ f ( y ) λ for every x , y ∈ R n , λ ∈ [0 , 1]. • F ( R n ) log-concave integrable functions in R n . • If f ∈ F ( R n ) ∩ W 1 , 1 then � � x � Π ∗ ( f ) = 2 | x | x ⊥ P x ⊥ f ( y ) dy , where P x ⊥ f ( y ) = max s ∈ R f ( y + sx ).

Functional affine inequalities Theorem 6 (Alonso-Guti´ errez, Bernu´ es, G.M. +2018) Let f ∈ F ( R n ). Then � � R n min { f ( y ) , f ( x ) } dydx ≤ 2 n n ! � f � n +1 | Π ∗ ( f ) | . 1 R n � f � ∞ = e −� x � △ for a simplex △ ∋ 0. f ( x ) ”=” iff

Functional affine inequalities Theorem 6 (Alonso-Guti´ errez, Bernu´ es, G.M. +2018) Let f ∈ F ( R n ). Then � � R n min { f ( y ) , f ( x ) } dydx ≤ 2 n n ! � f � n +1 | Π ∗ ( f ) | . 1 R n � f � ∞ = e −� x � △ for a simplex △ ∋ 0. f ( x ) ”=” iff Remark If f ( x ) = e −� x � K with K ∈ K n then Thm. 6 becomes Thm. 4, i.e. � 2 n � ≤ | K | n − 1 | Π ∗ ( K ) | . n n n

Proof of Theorem 6 Definition Let f ∈ F ( R n ). Then K t ( f ) := { x ∈ R n : f ( x ) ≥ e − t � f � ∞ } ∀ t ≥ 0 .

Proof of Theorem 6 Definition Let f ∈ F ( R n ). Then K t ( f ) := { x ∈ R n : f ( x ) ≥ e − t � f � ∞ } ∀ t ≥ 0 . Lemma 1 Let f ∈ F ( R n ). The covariogram g : R n → R of f � ∞ e − t | K t ( f ) ∩ ( x + K t ( f )) | dt g ( x ) := 0 � f ( y ) � , f ( y − x ) � = R n min dy � f � ∞ � f � ∞ is even and g ∈ F ( R n ).

Proof of Theorem 6 Lemma 2 Let f ∈ F ( R n ) and g its covariogram. For every 0 < λ 0 < 1 then K − log(1 − λ ) ( g ) � 2 � f � 1 Π ∗ ( f ) = . λ 0 <λ<λ 0

Proof of Theorem 6 Proof. Since K − log(1 − λ ) ( g ) /λ rewrites as

Proof of Theorem 6 Proof. Since K − log(1 − λ ) ( g ) /λ rewrites as � ∞ � ∞ � � x ∈ R n : e − t | K t ∩ ( λ x + K t ) | dt ≥ (1 − λ ) e − t | K t | dt = 0 0

Proof of Theorem 6 Proof. Since K − log(1 − λ ) ( g ) /λ rewrites as � ∞ � ∞ � � x ∈ R n : e − t | K t ∩ ( λ x + K t ) | dt ≥ (1 − λ ) e − t | K t | dt = 0 0 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x � � | x | + K t ) | x ∈ R n : e − t | K t | dt dt ≤ . λ 0 0

Proof of Theorem 6 Proof. Since K − log(1 − λ ) ( g ) /λ rewrites as � ∞ � ∞ � � x ∈ R n : e − t | K t ∩ ( λ x + K t ) | dt ≥ (1 − λ ) e − t | K t | dt = 0 0 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x � � | x | + K t ) | x ∈ R n : e − t | K t | dt dt ≤ . λ 0 0 Since | K t | − | K t ∩ ( λ | x | x | x | + K t ) | ≤ λ | x || P x ⊥ K t |

Proof of Theorem 6 Proof. Since K − log(1 − λ ) ( g ) /λ rewrites as � ∞ � ∞ � � x ∈ R n : e − t | K t ∩ ( λ x + K t ) | dt ≥ (1 − λ ) e − t | K t | dt = 0 0 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x � � | x | + K t ) | x ∈ R n : e − t | K t | dt dt ≤ . λ 0 0 Since | K t | − | K t ∩ ( λ | x | x | x | + K t ) | ≤ λ | x || P x ⊥ K t | λ x + K t λ | x | K t λ | x || P x ⊥ K t | | K t | − | K t ∩ ( λ x + K t ) |

Proof of Theorem 6 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x | x | + K t ) | e − t | P x ⊥ K t | dt dt ≤ | x | λ 0 0

Proof of Theorem 6 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x | x | + K t ) | e − t | P x ⊥ K t | dt dt ≤ | x | λ 0 0 = � x � Π ∗ ( f ) . 2 � f � ∞

Proof of Theorem 6 � ∞ � ∞ e − t | K t | − | K t ∩ ( λ | x | x | x | + K t ) | e − t | P x ⊥ K t | dt dt ≤ | x | λ 0 0 = � x � Π ∗ ( f ) . 2 � f � ∞ Therefore if � ∞ � e − t | K t | dt = 2 � x � Π ∗ ( f ) ≤ 2 � f � ∞ R n f ( x ) dx 0