On orthogonal m-pods on a cone

Abstract

Let S={ a 1≥ a 2≥…≥ a m } be a sequence of real numbers. If a i ≥0, a j ≤0 then the replacement of a i , a j by a i + a j will be called a merger. The set S can be transformed into a set S 1 of s≤ m real numbers by a sequence of mergers if and only if ∑ 1 s a j ⩾ ∑ 1 m a j ⩾ ∑ m − s + 1 m a j Applying this result the maximum number of orthogonal lines which can be inscribed into a cone is computed. In this paper I prove three theorems. Theorem 1 has been previously derived by M. Marcus [2] applying results of Ky Fan [3] but an independent constructive proof of Theorem 1 might be of interest. Theorem 2 is a consequence of Theorem 1 which has previously been stated only for n=3. Theorem 3 is a combinatorial theorem and is new as far as I can ascertain.