On the Homothety Conjecture ∗
نویسندگان
چکیده
Let K be a convex body in R and δ > 0. The homothety conjecture asks: Does Kδ = cK imply that K is an ellipsoid? Here Kδ is the (convex) floating body and c is a constant depending on δ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies B p , 1 ≤ p ≤ ∞, the unit balls of l p ; namely, we show that (B p )δ = cB p if and only if p = 2. We also show that the homothety conjecture is true for a general convex body K if δ is small enough. This improvs earlier results by Schütt and Werner [16] and Stancu [20].
منابع مشابه
Bilinear Forms on Frobenius Algebras
We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius k-algebra R with residue field k. If R is symmetric, then there exists a unique form on R up to homothety iff R is commutative. If R is Frobenius, then we introduce a norm based on the Nakayama automorphism of R. We show that if two forms on R are homothetic, then the norm of th...
متن کاملAddition and Subtraction of Homothety Classes of Convex Sets
Let SH denote the homothety class generated by a convex set S ⊂ R: SH = {a + λS | a ∈ R, λ > 0}. We determine conditions for the Minkowski sum BH + CH or the Minkowski difference BH ∼ CH of homothety classes BH and CH generated by closed convex sets B,C ⊂ R to lie in a homothety class generated by a closed convex set (more generally, in the union of countably many homothety classes generated by...
متن کاملOrigin–symmetric Bodies of Revolution with Minimal Mahler Volume in R3 –a New Proof
In [22], Meyer and Reisner proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R 3 , cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R3 , 3-cubes have the minimal Mahler volume. Mathematics subject classification (2010): 52A10, 52A40.
متن کاملA note on Fouquet-Vanherpe’s question and Fulkerson conjecture
The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe as...
متن کاملOn some generalisations of Brown's conjecture
Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|
متن کامل