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Cortona Some remarks on the girth functions of convex bodies Paolo Gronchi Dipartimento di Matematica e Applicazioni per lArchitettura Università degli Studi di Firenze (joint work with Stefano Campi)

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Cortona u What is the girth function? uK g K (u) = perimeter (K | u ) / g K (u) = perimeter (K | u ) / u u g K (u) = mean width (K | u )

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Cortona First property of the girth function g K (u) is a support function (when extended homogeneously) convex unit ball segment [u,-u] area measure of order 1

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Cortona g K (u) is the support function of Π 1 K, the projection body of order 1 of K Π 1 K is a zonoid

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Cortona Not all zonoids are projection bodies of order 1. K = segment parallel to uΠ 1 K = disk in u All Minkowski sums of disks are projections bodies of order 1.

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Cortona This suggests that h Π K (u) is not independent of its values on u. its values on u.1 In dimension 3, such an ellipsoid cannot be the sum of disks. It is easy to prove that, for any K, In higher dimensions

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Cortona This suggests that h Π K (u) is not independent of its values on u. its values on u.1 Integrating last inequality we find

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Cortona is equivalent to the existence of a box Psuch that

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9 The existence of a cylinder Csuch that uis equivalent to the statements as a function of v u v u is a support function. negative number!

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Cortona

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Cortona

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