Tomilenko, Vladimir A.

English

Tomilenko Vladimir A.

University of Szeged

Acta Sei. Math., 38 (1976), 417—341' On a property of strictly logarithmic concave functions V. A. TOMILENKO 1 . Introduction. In the work [1] by A . PREKOPA the following theorem was proved. ' T h e o r e m 1. Let f i x , y ) be a function of n+m variables, where x is an n-compo-nent and y is an m-component vector. Suppose that f is logarithmic concave in Rn+m and let Abe a convex subset of Rm. Then the function I(x)= ¡ f i x , y)dy A is logarithmic concave in the entire space R". The main result of this work is a similar statement for strictly logarithmic concave functions. Let / be a non-negative logarithmic concave function in Rn+m. We denote D = {z£Rn+m: f ( z ) > 0}, D(x) = {y£Rm: f ( x , y) > 0 } , B = : I(x) > 0} . The sets D(x) (x£Rn), D and B are convex in Rm, R"+m and R", respectively. The relative interior of a convex set C c Rk is denoted by riC (see [2] p. 57) and the closure of C by C. The basic theorem of this work is T h e o r e m 2. Let f i x , y ) be a function of n+m variables where x£R", y£Rm. Suppose f is logarithmic concave in Rn+m and strictly logarithmic concave in ri D, and let A be convex subset of the space Rm. If the sets Z ) ( x ) c i ? m are bounded for every x£R", then the function I is logarithmic concave in the entire space R" and strictly logarithmic concave in ri B. The first part of this statement is just Theorem 1. We shall begin with proving the strictly logarithmic concavity of the function I in tiB with subsidiary statements. In this work the terminology has been taken from [2]. 2. Auxiliary statements. We define the function g:i?"+m—R as follows g(z) = - l n / ( z ) , z = (x, y)ZR»+m. Received June 13, 1975. 418 V. A. Tomilenko Under the conditions imposed on f g is a proper convex function with effective domain d o m g = {z£R"+m : g(z) < = £>. We denote / * ( z ) = lim s u p / 0 0 , v, z£Rn+m. V — Z Lemma 1. For all z£Rn+m (cl*)(z) = - l n / , ( z ) , where cl g is the closure of the convex function g. Proof . From the definition of cl g ([2] p. 67—68) and g we have (cl g)(z) = lim inf g(v) = lim inf [— In/(«)] = —lim sup In f(v). . v—z V—Z V—Z The continuity and strict monotonicity of the logarithm implies that lim sup In f(v) = In [lim sup /(«)] = In /*(z). v—z V—Z The lemma is proved. C o r o l l a r y 1. The function is logarithmic concave in Rn+m. C o r o l l a r y 2. The function f agrees with in Rn+m except perhaps at relative boundary points of a convex set D. Corollaries 1 and 2 follow from Theorem 7.4 [2] and Lemma 1. L e m m a 2. I f f is upper semi-continuous on the closed bounded set DaRk, then there exists z0£D such that sup / ( z ) = / ( z 0 ) . ziD Proof. Let s u p / ( z ) = C and e„>0, s„—0 as « — T h e n one can find a se-z£D quence {z„}cD such that for « = 1, 2, . . . / ( z n ) > C - 6 „ . Since D is a bounded closed set without loss of generality we may assume that z„ — z0 as n-—°o, z 0 £ A and |z„—z0| ^ e„ for « = 1 ,2 , . . . . Hence the inequality (1) sup . / ( z ) i=/(z„) > C-EN, « = 1 , 2 , ... | z 0 -z | -=c„ is valid. Taking into account the upper semi-continuity of the function / we get from (1) that / (z 0 ) = lim sup / ( z ) s C. Thus/(z0)=C. The lemma is proved. On a property of strictly logarithmic concave functions 419 L e m m a 3. Let z1,z2£Rn+m and 0 < A < 1. I f f is strictly logarithmic concave in ri D<zRn+m and Xz1+(\—X)z2^riD, then the inequality (2) / * ( A z i + ( l ~X)z2) > f t ( z J H z , ) is valid. Proof . Two cases are possible. (i) One of the points, either zl or z2, does not belong to D. In this case inequal-ity (2) is obviously correct. (ii) Let zlt Z2£D. Let us draw a straight line I across the points z1 and z2 and choose some point zfJC\n D. Let (p(/i)=g(/xz1+(l —fi)z2). Then cl <p is a proper strictly convex function on [0, 1]. From Theorems 7.4 and 7.5 of [2] it follows that (cl cp) (ji)=(p(ji) f o r (0, 1) a n d (cl<p)(l) = lim(v + ( l -v ) / i 0 ) = limg(vz1 + ( l - v ) z ) = (cl g)(zl), (cl q>)(0) = lim (/i0-v/x0) = lim g(vz2 + ( 1 -v )z ) = (cl g)(z2), vtl vtl N 'where z=n0z1 + (l~n0)z2. This means that the function cl g is strictly convex on the set l f l D, that is (3) (cl g ) ( l Z l + ( l — A)Z2) < ;.(cl g)(z!)+(l —A)(cl g)(z2), 0 < A < 1. From (3) and Lemma 1 it can be seen is proved. that inequality (2) is true. The lemma C o r o l l a r y 3. Let zlt z2£R"+m and 0 < A < 1. If f is strictly logarithmic concave in ri D(ZRn+m and Azx + (1 — A ) z 2 £ r i D, then we have the inequality /(Az1 + (1 —A)z2) ( z i ) / 1 " ^ ) . Q L e m m a 4. If x0 B, yu(i\nt D (x 0) , then z 0 —(x0i)>0)(;riZ). Proof. Let P be the projection (x, y)-*x from Rn+m onto R". It can be shown that BczPD and if B is not empty then the dimension of the set B agrees with that of PD. Hence ri 5 c ri (PD) and the point (x0, j>0) 6 ri D by Theorem 6.8 of [2]. The lemma is proved. 3. Proof of Theorem 2. We denote D^x) = { y £ E r : f t ( x , - y ) > 0 } . For all x£ri B the sets D(x) and D^(x) have the same closure and the same interior (see Corollary 2). 420 V. A. Tomilenko Let , x2 6 ri B, 0 < A < 1 and x0=Xxi +(1 — X)x2. We define the functions / i and f2 as follows: /iOO =/*(*i> J) if and fi(y) = 0 otherwise; / 2 0 0 =f*(*2, >0 if y£A, and f 2 ( y ) = 0 otherwise . For given y£Rm and A, 0 < 2 < 1 , we shall denote by <S(y; A) the set of points («, v) such that u, v£Rm, ku+(l-X)v=y. It can be shown that for all y£Rm s u p f * ( X l , u)f}~\x2, v) S s u p f\(u)f2~x(v) Sir,*) S(y;X) and f o r y£Af]D(x0) s u p f H u ) f i ~ \ v ) = 0 . SG>;A) Since is logarithmic concave in Rn+m (Corollary 1), the following inequality will be valid for all y£Rm: / * (*o> y) S s u p f£(Xj, u)fl~x(x2,v). We shall prove that for all j6 intD(x 0 ) we have (4) f*(x0, >>) > s u p f f ( x i , u)fi~x(x2, v). Suppose on the contrary that there could be found a y0€int D(x0) such that /* (*<» Jo) = s u p f*(xi,u)f^~x(x2,v). In this c a s e ( x 0 , y0)>0 as (x„, j0)€ri D (Lemma 4). According to Lemma 2 there exists a point (u0, v0)€S(y0; X) such that w 0 e £ ( * i ) > a n d f j x 0 , y0) =f*(x1, w 0 ) / * 1 - ; ( * 2 , v0). We have got a contradiction to Lemma 3. So, for all j>£int D(x0) inequality (4) is valid. From the definition of the function / and from Corollary 2 we get / ( * „ ) = / / ( * o . y ) d y = / / • ( * o , y)dy. A A n B(xq) On a property of strictly logarithmic concave functions 421 Taking into account (4) and Theorem 3 of [1] we obtain: J f*(xo> y)dy > / s u p f£(Xi, u)f*~'{x2, v)dy ë — / s u p f h u ) f t x { v ) d y = / s u p tf(u)ftx(v)dyë ^ I f A M d y Y l f A W d y y - ^ l f f ¥ ( x 1 , y ) d y ] l [ f U x 2 , y ) d y ] 1 - ' = Rm Rm iflDI*,) • ADBixd = [I(xù№xùï-X' The theorem is proved. Corol lary 4. Let x1,x2£Rn and 0<A<1 . If /bcj+O—A)x26ri5, then the inequality (5) l{Xx1 + (\-X)x2) > [ / ( ^ [ / ( x , ) ] 1 ^ is valid. i Proof . It follows from Theorem 2 and Corollary 3. In conclusion the author expresses his gratitude to G. G. Pestov for his help in carrying out the present work. References [1] A. PREKOPA, On logarithmic concave measures and functions, Acta Sci. Math., 34 (1973), 335-343. [2] P. Р о к а ф е л л а р , Выпуклый анализ, Изд-во Мир (Москва, 1973). CHAIR OF MATH. ANALYSIS UNIVERSITY TOMSK TOMSK 634010, USSR 

On a property of strictly logarithmic concave functions

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On a property of strictly logarithmic concave functions

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