Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of this cell-decomposition, and for each of them define $A_k$ to be the $k$-dimensional volume. Now, the question is, are there good (whatever that means) lower and upper bounds for the $A_k$s, given a bound on, say, $A_0?$
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asked Apr 3, 2015 at 15:15
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You can get some bounds using Crofton formula. It semms that the optimal bound is $$A_k\ge \tfrac12\cdot(n-k+2)\cdot\mathrm{vol}_k\mathbb S^k$$
The lower bound on $A_k$ does not seem to change when you increase $A_0$.
answered Apr 3, 2015 at 17:58
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$\begingroup$ Presumably the lower bound does not increase because vertices can coalesce, so we are down to the case of the simplex?! $\endgroup$ Commented Apr 3, 2015 at 19:28
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$\begingroup$ Yes, and hopefully the formula is correct. $\endgroup$ Commented Apr 3, 2015 at 21:03