Vol. 56, No. 1, 1975

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The orthocentric simplex as an extreme simplex

Leon Gerber

Vol. 56 (1975), No. 1, 97–111

Let 𝒜 be a variable n-simplex containing a fixed point Q and having vertices Ai, and corresponding opposite faces 𝒜i,i = 0,1,,n. We use the properties of orthocentric simplexes to present brief solutions to the following problems and obtain several Erdös-Mordell type inequalities as a by-product, some of which are stronger than known inequalities.

(i) Maximize the volume 𝒜 given the distances QAi = dl 0,i = 0,,n.

(ii) Minimize the volume 𝒜 given the distances e,0 from Q to 𝒜i,i = 0,,n.

(iii) Find the extrema of (i) and (ii) when only the power means of the distances are given.

(iv) Construct an orthocentric simplex given the lengths of the altitudes.

(v) Maximize the volume of 𝒜 given the (n 1)-dimensional volumes of the faces.

(vi) Find the maximum in (i) given that Q must be the centroid of 𝒜.

(vii) Maximize the volume of the convex hull of a skew (n + 1)-gon given the power means of its edges.

Mathematical Subject Classification 2000
Primary: 10E10
Secondary: 52A40
Received: 2 August 1973
Published: 1 January 1975
Leon Gerber