Download Art and Geometry: A Study in Space Intuitions (Dover Books by William M. Ivins PDF

By William M. Ivins

A hugely stimulating research of the occasionally startling interrelationships among artwork and arithmetic all through background, as saw via the traditional Greeks, Renaissance theorists, artists corresponding to Albrecht D?rer, etc.

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Extra resources for Art and Geometry: A Study in Space Intuitions (Dover Books on Art History)

Example text

This step is carried over by means of approximation by solutions of the homogeneous problem (Lemma 11). A variation of Lemma 2 is the following: 24 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION Lemma 9. Let u be an element of S(f) in fl D B1. Assume that DI nQr, (x1) 36 0 and that Ilf II L"(B,) 5 8 (small enough), then there exist M and u, depending on A,,\, r1i r2,17, such that IDM nQr,(xi)I > 1z1Qr,(x1)I > 0. Proof. Let xo E D1 n Qr, (x1). Subtracting from u a linear function, we may suppose that the paraboloid at xo is 1-1x12.

Corollary 2. Assume that u is strictly convex, then u has a unique supporting plane at each point. Proof. If not we may assume that a) axn (a>0), b) u(0) = 0, c) u(-ten) = 0(t) (t > 0). We now consider the auxiliary function tl,,r =u-T(Xn+a). From the strict convexity of u, the set u,,r < 0 becomes compactly contained in the domain of definition of u for a, T positive and small. Also for T < a, u,,r attains its minimum at x = 0. Finally, we estimate the location of the two supporting planes to the set u,,r <0 of the form HI = {xn = C-},113 = {xn = C+}.

Indeed, consider a sequence uk for which sup h112(X - Xo) > 1 - 1/k. x#xo hl (X - Xo) From Lemma 1, Xo, {uk = 1/21 and {uk = 1} stay uniformly away from each other. In particular 0 < CI < C2. C1 IX -X01