The width w of a curve γ in Euclidean space Rn is the infimum of the distances between all pairs of parallel hyperplanes which bound γ , while its inradiusr is the supremum of the radii of all spheres which are contained in the convex hull of γ and are disjoint from γ . We use a mixture of topological and integral geometric techniques, including an application of Borsuk Ulam theorem due to Wienholtz and Crofton’s formulas, to obtain lower bounds on the length of γ subject to constraints on r and w. The special case of closed curves is also considered in each category. Our estimates confirm some conjectures of Zalgaller up to 99% of their stated value, while we also disprove one of them

EB00000003420K

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