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Euclidean projections of a p-convex body

Part of the Lecture Notes in Mathematics book series (LNM,volume 1745)

Abstract

In this paper we study Euclidean projections of a p-convex body in ℝn. Precisely, we prove that for any integer k satisfying ln nkn/2, there exists a projection of rank k with the distance to the Euclidean ball not exceeding C p (k/ln(1 + n/k))1/p−1/2, where C p is an absolute positive constant depending only on p. Moreover, we obtain precise estimates of entropy numbers of identity operator acting between ℓ p and ℓ r spaces for the case 0 < p < r ≤ ∞. This allows us to get a good approximation for the volume of p-convex hull of n points in ℝk, p < 1, which shows the sharpness of the announced result.

Keywords

  • Convex Hull
  • Convex Body
  • Euclidean Ball
  • Convex Case
  • Symmetric Body

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The second named author holds the Lady Davis Fellowship.

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Guédon, O., Litvak, A.E. (2000). Euclidean projections of a p-convex body. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0107210

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  • DOI: https://doi.org/10.1007/BFb0107210

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