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Radial Minkowski additive operators

Lewen Ji (2021)

Czechoslovak Mathematical Journal

We give some characterizations for radial Minkowski additive operators and prove a new characterization of balls. Finally, we show the property of radial Minkowski homomorphism.

Random ε-nets and embeddings in ℓ ∞ N

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

We show that, given an n-dimensional normed space X, a sequence of N = ( 8 / ε ) 2 n independent random vectors ( X i ) i = 1 N , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map Γ : ℝ → ℝ N defined by Γ x = ( ⟨ x , X i ⟩ ) i = 1 N embeds X in ℓ ∞ N with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ℓ ∞ N with asymptotically best possible relation between N, n, and ε.

Reduced spherical polygons

Marek Lassak (2015)

Colloquium Mathematicae

For every hemisphere K supporting a spherically convex body C of the d-dimensional sphere S d we consider the width of C determined by K. By the thickness Δ(C) of C we mean the minimum of the widths of C over all supporting hemispheres K of C. A spherically convex body R ⊂ S d is said to be reduced provided Δ(Z) < Δ(R) for every spherically convex body Z ⊂ R different from R. We characterize reduced spherical polygons on S². We show that every reduced spherical polygon is of thickness at most π/2. We...

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