Let [itex]\mathbb{S}^n[/itex] be a simplex in [itex]\mathbb{R}^{n+1}[/itex], so [itex]\mathbb{S}^{n}=\{x\in\mathbb{R}^{n+1}|\sum{}x_{i}=1\}[/itex]. Let [itex]D[/itex] be a difference measure on [itex]\mathbb{S}^{n}[/itex] with [itex]D(x,x)=0[/itex] and [itex]x=y[/itex] for [itex]D(x,y)=0[/itex]. [itex]D[/itex] is also smooth, so differentiable as much as we need.(adsbygoogle = window.adsbygoogle || []).push({});

Let(R)be a convexity requirement for [itex]D[/itex], corresponding to the intuition that on the periphery points are farther apart from each other than near the centre. [itex]M[/itex] is the midpoint of [itex]\mathbb{S}^{n}[/itex], so the coordinates of [itex]M[/itex] are all [itex]m_{i}=1/(n+1)[/itex]. Let [itex]d[/itex] be the Euclidean distance function.

(R)if conditions (i)--(iii) are fulfilled for [itex]X_{1},X_{2},X_{3},X_{4}[/itex] then [itex]D(X_{1},X_{2})^{2}>D(X_{3},X_{4})^{2}[/itex].

Here are conditions (i)--(iii) (see the diagram for illustration in [itex]\mathbb{S}^{1}[/itex]):

http://streetgreek.com/lpublic/various/horizon.png [Broken]

(i) [itex]X_{1},X_{2},X_{3},X_{4}[/itex] are all on a straight line between [itex]M[/itex] and an arbitrary point [itex]Z[/itex] on the boundary of [itex]\mathbb{S}^{n}[/itex], so [itex]v(OX_{i})=v(OM)+\lambda_{i}v(MZ)[/itex] (where [itex]v(OX)[/itex] is the vector from the origin to [itex]X[/itex] and [itex]\lambda_{i}\in[0,1][/itex]).

(ii) [itex]d(X_{1},X_{2})=d(X_{3},X_{4})[/itex].

(iii) [itex]\max\{\lambda_{i}\}=\lambda_{2}[/itex], so the pair [itex]X_{1},X_{2}[/itex] is farther away from the midpoint than the pair [itex]X_{3},X_{4}[/itex].

I am using the squared difference in(R)because I am allowing [itex]D[/itex] to be negative-valued. Given only(R)can I show that [itex]M[/itex] is the only stationary point on [itex]\mathbb{S}^{n}[/itex] for [itex]D[/itex] such that

$$

\frac{\partial{}D}{\partial{}x_{i}}=0

$$

only at [itex]M[/itex]? The idea would be to show that a non-zero partial derivative at [itex]M[/itex] is inconsistent with(R)and a zero-vector derivative anywhere else is inconsistent with(R)as well. I lack the mathematical sophistication for a proof. I don't even really know what methods to use and which tags to give to this post.

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# Stationary point for convex difference measure

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