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(A) Closed and bounded centrally symmetric sets S in E n can also be characterized by the property that for each n-dimensional simplex T with vertices in S there is a translate of −T also having its vertices in S. This, of course, is a consequence of Theorem 1, since the three-point sets belonging to S are subsets of the (n + 1)-point sets with vertices in… (More)

An O(n) test for polygon convexity is stated and proved. It is also proved that the test is minimal in a certain exact sense. Everyone knows a convex polygon when one sees it. Yet, to deal with the notion of polygon convexity mathematically or computationally, it must be adequately described. A convex polygon can be deened, as e.g. in [6, page 5], as a… (More)

We provide an exact asymptotic lower bound on the mini-max expected excess risk (EER) in the agnostic probably-approximately-correct (PAC) machine learning classification model. This bound is of the simple form c∞/ √ ν as ν → ∞, where c∞ = 0.16997. .. is a universal constant, ν = m/d, m is the size of the training sample, and d is the Vapnik–Chervonenkis… (More)

The main results imply that the probability P(Z ∈ A + θ) is Schur-concave/Schur-convex in (θ 2 1 ,. .. , θ 2 k) provided that the indicator function of a set A in R k is so, respectively; here, θ = (θ 1 ,. .. , θ k) ∈ R k and Z is a standard normal random vector in R k. Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set… (More)

An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges [V_0,V_1],...,[V_{n-1},V_0]; if at that no three vertices of \P are collinear then \P is called strictly convex. We prove… (More)