REVERSES OF THE TRIANGLE INEQUALITY IN INNER PRODUCT SPACES

We show that if x(1), ... , x(n) are vectors in a normed linear space (X, vertical bar vertical bar center dot vertical bar vertical bar) and s1, ... , s(n) belong to the interval [0, infinity), then f(n)(s(1), ... , s(n)) = Sigma(n, j = 1) vertical bar vertical bar s(j)x(j)vertical bar vertical bar - vertical bar vertical bar Sigma(n, j = 1) s(j)x(j)vertical bar vertical bar is a non-negative valued continuous function such that f(n)(s(1), ... , s(n)) <= f(n)(t(1), ... , t(n)) for all s(1), ... , s(n) and t(1), ... , t(n) in [0,infinity) with s(j) <= t(j) (1 <= j <= n). By using it, we prove several versions of reverse triangle inequality in inner product spaces and discuss equality attainedness of norm inequalities in strictly convex Banach spaces.