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Floating body, illumination body, and polytopal approximation. (English. Abridged French version) Zbl 0884.52007
This paper generalizes some results, obtained in the past by several authors, concerning the approximation of convex bodies by means of polytopes.
The author and E. Werner have defined elsewhere [Math. Scand. 66, No. 2, 275-290 (1990; Zbl 0739.52008)] the convex floating body $$K_t$$ of a convex body $$K$$ as the intersection of all halfspaces whose defining hyperplanes cut off a set of volume $$t$$ from K. E. Werner has introduced [Stud. Math. 110, No. 3, 257-269 (1994; Zbl 0813.52007)] the illumination body $$K^t$$ of a convex body $$K$$ as the set $$\{x\in R^d \mid \text{vol}_d ([x,K] \backslash K) \leq t\}$$.
The author then states the following results:
Theorem 1: For any convex body $$K$$ in $$\mathbb{R}^d$$ there exists an $$n\in N$$ such that $n\leq e^{16d} \biggl\{\bigl [\text{vol}_d (K \setminus K_t)\bigr]/ \bigl[t \text{vol}_0 (B^d_2)\bigr] \biggr\}$ for $$\forall t$$ satisfying the condition $$0\leq t\leq(1/4) c^{-4} \text{vol}_dK$$ there exists a polytope $$P_n$$ with $$n$$ vertices such that $$K_t \subset P_n \subset K$$.
Theorem 2: Let $$K \subset \mathbb{R}^d$$ be a convex body such that $$(1/c_1) B^d_2\subset K\subset c_2 B^d_2$$ let $$t$$ satisfy the condition $$0\leq t \leq (5c_1c_2)^{-d-1} \text{vol}_dK$$ and $$n\in N$$ be such that $(128 \pi/7)^{(d-2)/2} \leq n\leq (32 edt)^{-1} \text{vol}_d (K^t \setminus K)$ Then $\text{vol}_d(K^t \setminus K) \leq 10^7 d^2(c_1 c_2)^{2+ [l/(d-1)]} \text{vol}_d (P_n \setminus K)$ for every polytope $$P_n$$ containing $$K$$ with at most $$(d-1)$$-dimensional faces.
The proof of this theorem will be given elsewhere.

##### MSC:
 52A27 Approximation by convex sets
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