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Let $ℱ = F_{α}$ be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly’s theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = maxn+1,2n-2d+2 for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski’s result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball of measure...

Minimal fixing systems for convex bodies.

Boltyanski, V.G., Morales Amaya, E. (1995)

Journal of Applied Analysis

Minimal sets of vectors which generate $R_{n}$ with excess $k$

Miroslav Fiedler (1979)

Czechoslovak Mathematical Journal

Minkowski valuations intertwining the special linear group

Christoph Haberl (2012)

Journal of the European Mathematical Society

All continuous Minkowski valuations which are compatible with the special linear group are completely classiﬁed. One consequence of these classiﬁcations is a new characterization of the projection body operator.

More on exposed points and extremal points of convex sets in $ℝ^{n}$ and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

Let $𝕍$ be a separable real Hilbert space, $k \in ℕ$ with $k < dim 𝕍$ , and let $B$ be convex and closed in $𝕍$ . Let $𝒫$ be a collection of linear $k$ -subspaces of $𝕍$ . A point $w \in B$ is called exposed by $𝒫$ if there is a $P \in 𝒫$ so that $(w + P) \cap B = {w}$ . We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by $𝒫$ points whenever $𝒫$ is dense and $G_{δ}$ . In addition, we discuss the question when the set of exposed by some $𝒫$ points forms a $G_{δ}$ -set.

Multiapplications boréliennes à valeurs convexes de $ℝ^{n}$

Jean Saint-Pierre (1976)

Annales de l'I.H.P. Probabilités et statistiques

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