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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.
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