Coverings by convex bodies and inscribed balls

الأصل · EN

Let H be a Hilbert space. For a closed convex body A denote by r(A) the supremum of radiuses of balls, contained in A. We prove, that ∑ₙ₌₁∞ r(Aₙ) ≥ r(A) for every covering of a convex closed body A ⊂ H by a sequence of convex closed bodies Aₙ, n ∈. It looks like this fact is new even for triangles in a 2-dimensional space.