Daniel M. Oberlin Department of Mathematics, Florida State University. May 2004

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1 THE SECOND DERIVATIVE OF A CONVEX FNCTION Daniel M. Oberlin Department of Mathematics, Florida State niversity May 004 Suppose f is a convex function on an open interval I. The following facts are well known and easy to verify: (a the second (distributional derivative of f is a nonnegative locally finite Borel measure on I, and any such measure is the second derivative of a convex function f which is unique up to the addition of an affine function; (b it follows from (a that (1 f, φ 0 whenever φ Cc (I is nonnegative; conversely, if f is a distribution on I which satisfies (1, then f is a convex function. The purpose of this note is to describe certain analogues of these facts for functions of several variables and then to prove Theorem below. Thus let be a convex open subset of R n. Suppose f is a convex function on. Then, as a continuous function, f defines a distribution on. The derivatives of f mentioned below are to be interpreted in that sense. In particular, the second derivative D f is the Hessian matrix of distributions on whose entries are the second-order partial derivatives f xix j of f. Let B 0 ( be the collection of all Borel sets E having compact closures contained in. An analogue of (a is Theorem 1. Suppose f is convex on. For 1 i, j n there are real-valued set functions µ ij : B 0 ( C which are complex measures on any compact subset of. For all φ C 0 ( and k = 1,, n, the µ ij satisfy the equations ( f xix j, φ = µ ij, φ, (3 µ ij, φ xk = µ ik, φ xj. Further, the symmetric matrix µ(e = ( µ ij (E is positive definite for all E B 0 (. Conversely, if µ = (µ ij 1 Typeset by AMS-TEX

2 DANIEL M. OBERLIN is such a (positive definite matrix valued set function, then there is a convex function f on with D f = µ. Such an f is unique up to the addition of an affine function. The following analogue of (b is a corollary of Theorem 1. Corollary. Suppose f is a distribution on satisfying f xix j, φ i φ j 0 whenever φ i C 0 (. Then f is a convex function on. More interestingly, there is a lower bound on the Hausdorff dimension of the support of D f for nontrivial convex functions f: Theorem. With D f = µ as above, if µ is supported on a Borel set having Hausdorff dimension less than n 1, then f is affine. To begin the proofs, suppose that η is a nonnegative and radial C function supported in B(0, 1 and having integral 1. With η ɛ (x = ɛ n η(x/n for ɛ > 0 and for ρ a distribution on, let ρ ɛ = η ɛ ρ be the usual ɛ-mollification of ρ, a distribution acting on those φ C 0 ( whose support is distant at least ɛ from. Proof of Theorem 1: Suppose that f is convex on. Fix an open V such that V has compact closure contained in and choose ɛ 0 > 0 such that V + B(0, 3ɛ 0. Then if 0 < ɛ < ɛ 0, f ɛ is convex and C on V. Suppose ψ C0 is nonnegative, identically 1 on V, and supported in V + B(0, ɛ 0. Since 0 as ɛ 0, the nonnegative measures V fx ɛ ix i (x dx fx ɛ ix i (xψ(xdx = f ɛ (xψ xix i (x dx f(xψ xix i (x dx χ V (xf ɛ x ix i (x dx have uniformly bounded total variations on V. Since fx ɛ ix j fx ɛ ix i 1/ fx ɛ jx j 1/ f x ɛ ix i + fx ɛ jx j, the same is true for the complex measures χ V (xf ɛ x ix j (x dx. Taking weak* its gives complex measures µ ij on V satisfying ( and (3 if φ C0 (V. Letting V then yields the set functions µ ij whose existence is

3 THE SECOND DERIVATIVE OF A CONVEX FNCTION 3 the first claim of Theorem. (The statement about the positive definite character of µ(e follows from V f ɛ x ix j (xφ i (xφ j (x dx 0 and a it argument. Suppose now that µ = ( µ ij is as in Theorem. Suppose that V and ɛ0 are as above and suppose additionally that V is C 1. Fix i, j = 1,..., n. For ɛ < ɛ 0, the C functions µ ɛ ij have uniformly bounded L1 (V + B(0, ɛ norms (since the set functions µ ij are complex measures on V + B(0, ɛ 0. The conditions (3 and the symmetry of µ imply that there are C functions f ɛ on V + B(0, ɛ 0 with f ɛ x i x j = µ ɛ ij. The Poincaré-Sobolev inequalities and the uniform bounds on µ ɛ ij L 1 (V +B(0,ɛ 0 show that there is q = q(n > 1 such that the functions f ɛ can be chosen to have f ɛ L q (V +B(0,ɛ 0 C for some positive C independent of ɛ. Passing to a subsequence which converges weakly in L q (V + B(0, ɛ 0 shows that there is f L q (V + B(0, ɛ 0 satisfying (4 f x i x j = µ ij (in the sense of distributions on V + B(0, ɛ 0. To check that f is equal a.e. to a convex function on V, we begin by noting that, for 0 < ɛ < ɛ 0, (4 and the positive definite assumption on µ show that the mollifications f ɛ are convex on V. The formula g (0 = 1 Σ n 1 0 r η(rσ r d dt g(tσrn 1 dr dσ + g(0 for the -mollification of a twice differentiable function g at 0 shows that -mollifications of twice differentiable convex functions decrease pointwise as 0. In particular, if B(x, V and 0 < < 0, this applies to (f ɛ (x for each ɛ and so, in the it as ɛ 0, to f (x. If 0 f (x = held for any x V, it would follow from convexity that f L 0 q (V. Thus the decreasing it 0 f (x

4 4 DANIEL M. OBERLIN gives a realization of f as a convex function on V. Now suppose that Ṽ satisfies the same hypotheses as V and is such that V Ṽ. Then the argument above yields a convex function f on Ṽ satisfying (4 on Ṽ. Such an f can be adjusted by the addition of an affine function to agree with f on V. Considering a sequence V 1 V of such open sets with V n = furnishes a convex f on satisfying D f = µ on. Since such an f is clearly unique up to the addition of an affine function, the proof of Theorem 1 is complete. Proof of Corollary: If f is a distribution on satisfying the hypothesis, then it is clear that each f xix i is nonnegative and therefore realizable as a nonnegative Borel measure µ ii on. The hypothesis also implies that f xix j, φ ζ i ζ j 0 for all real numbers ζ 1,..., ζ n whenever φ C 0 (. It follows that f xix j, φ f x ix i, φ + f xjx j, φ = µ ii, φ + µ jj, φ for φ C 0 (. This implies that each f xix j is realizable as a complex Borel measure µ ij on each open and bounded V. Thus the desired result follows from Theorem 1. Proof of Theorem : It follows from the proof of Theorem 1 that if f is convex on, each of the distributions f xj is realizable as a function on. Thus Theorem can be proved by applying the following result to each f xj : Lemma 1. Suppose the locally integrable function g on has the property that each of the distributions g xj is realized by a Borel measure µ j supported on a set of Hausdorff dimension less than n 1. Then g is equal a.e. to a constant function. The proof depends on a simple fact: Lemma. Suppose E R n is a set of Hausdorff dimension less than n 1. Then the n-dimensional Lebesgue measure of is 0. t>0 ( te Proof of Lemma : The mapping π : x x x is Lipschitz on E. = E { < x < 1/} for each > 0. Thus π(e has (n 1-dimensional measure 0 in Σ n 1 and the desired result follows by integrating in polar coordinates. Proof of Lemma 1: Assume without loss generality that 0 and that gɛ (0 = g(0.

5 THE SECOND DERIVATIVE OF A CONVEX FNCTION 5 This is possible since the local integrability of g implies that gɛ (x = g(x for a.e. x. If h is C 1 on and x, then leads to h(x = 1 0 x j h xj (xt dt + h(0 h(xφ(x dx = h(0 φ(x dx + for φ C 0 ( and = (φ > 0. Thus [ g ɛ (0 φ(x dx + Letting φ t (x = φ(x/t, we have uniformly for t 1 and so 1 g, φ = g ɛ, φ = 1 φ( x t n φ( x t n gx ɛ j (xx j φ t (x dx = g xj, x j φ t (5 g, φ = ( g ɛ (0 φ(x dx + 1 x j h xj (x dt dx tn+1 x j g ɛ x j (x dt t n+1 dx]. φ( x t n x j g xj (x dt dx. tn+1 If E is a set of Hausdorff dimension less than n 1 which supports each of the measures g xj, then (5 implies that g, φ = ( g ɛ (0 φ(x dx + φ(x dν(x where the measure ν is supported on the set t>0 ( te. By Lemma, that set has Lebesgue measure 0. Since g is a locally integrable function, it must be the case that ν = 0, and Lemma 1 follows.