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We use one-dimensional differential inequalities to estimate the squareness and type of Banach spaces with modulus of convexity of power type two. The estimates obtained are sharp and the constants involved moderate.
We consider the question whether the assumption of convexity
of the set involved in Clarke-Ledyaev inequality can be relaxed. In the case
when the point is outside the convex hull of the set we show that Clarke-Ledyaev
type inequality holds if and only if there is certain geometrical relation between the point and the set.
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