EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID
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For 0 < p ≤ ∞ and a convex body K in Rⁿ, Lutwak, Yang and Zhang dened the concept of dual Lp-centroid body T-pK and Lp- John ellipsoid EpK. In this paper, we prove the following two results:(i) For any origin-symmetric convex body K, there exist an ellipsoid E and a parallelotope P such that for 1 ≤ p ≤ 2 and 0 < q ∞,EqE ⊇ Γ-pK ⊇ (ncn-2p)-1/p EqP and V(E) = V(K) = V(P);For 2 ≤ p ≤ ∞ and 0 < q ≤ ∞,2-1 ωn1/n EqE ⊆ Γ-pK ⊆ 2ωn-1/n (ncn-2,p)-1/p EqP and V(E) = V(K) = V(P). (ii) For any convex body K whose John point is at the origin, there exists a simplex T such that for 1 ≤ p ≤ ∞ and 0 < q ≤ ∞,an(ncn-2,p)-1/p EqT ⊇ Γ-pK ⊇ (ncn-2,p)-1/p EqT and V(K) = V(T).
Abstract
For 0 < p ≤ ∞ and a convex body K in Rⁿ, Lutwak, Yang and Zhang dened the concept of dual Lp-centroid body T-pK and Lp- John ellipsoid EpK. In this paper, we prove the following two results:(i) For any origin-symmetric convex body K, there exist an ellipsoid E and a parallelotope P such that for 1 ≤ p ≤ 2 and 0 < q ∞,EqE ⊇ Γ-pK ⊇ (ncn-2p)-1/p EqP and V(E) = V(K) = V(P);For 2 ≤ p ≤ ∞ and 0 < q ≤ ∞,2-1 ωn1/n EqE ⊆ Γ-pK ⊆ 2ωn-1/n (ncn-2,p)-1/p EqP and V(E) = V(K) = V(P). (ii) For any convex body K whose John point is at the origin, there exists a simplex T such that for 1 ≤ p ≤ ∞ and 0 < q ≤ ∞,an(ncn-2,p)-1/p EqT ⊇ Γ-pK ⊇ (ncn-2,p)-1/p EqT and V(K) = V(T).
참고문헌 (17)
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