By Parshin Shafarevich

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**Example text**

U11)2 Finally, an approximation lemma on the image of the gradient mapping: Lemma 2. Let 6 > e > 0, and u, v two convex functions in Bl with Ilu-vliL. < e. Then (with N26 the 26-neighborhood) {VuIn1_,,,} C N26{VvlB,}. 50 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION Proof. Consider v` = v + e - 6(1 - r2). Then V*188, > u and v*IB,_C,a < U. Therefore any plane of support of u in BI-E/6 is a plane of support of v` in B1. That is {VUIB,_E,5 } C N26 {Vvla, } . Remark. (Pogorelov, see [P2].

Hence C+ < Ta On the other hand, for x = -te,,, we choose a small and set T au (note that T = 0(1) for or going to zero). For t < 2 , we get 2lv,r(-ten) = u(-ten) - 2u(2 aen) (-t + a) or (- 2 en) <0. Hence C- > -2' As in the proof of Theorem 1, we get a contradiction from the fact that goes to zero, by considering the three planes 111,113 and 112 = {x = 0}. PART 2. GEOMETRIC PROPERTIES OF THE MONGE AMPERE EQUATION 39 Corollary 3. Let u be a (convex) solution of 0

In this way the geometrical situation is the one described in Lemma 3. The proof is carried out in four steps: (I) To prove that if in some point in Q2, u is less than one, then u is bounded by a universal constant in a large portion of Ql, (Lemma 5). (II) Iteration of the result in (I) gives a polynomial decay for the distribution function of u in Q2, (Lemma 6). The corollary of the Calderon-Zygmund decomposition lemma is the key tool to connect each step in the iteration with the next one. (III) To prove that if u is large at a point well inside Q2, then u is larger at some point nearby (Lemma 7).