TY - JOUR
T1 - Algebraically inspired results on convex functions and bodies
AU - Rotem, Liran
N1 - Publisher Copyright:
© 2016 World Scientific Publishing Company.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body K° or the dual function φ∗ play the role of the inverses "K-1" and "φ-1", we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function φ one has (φ + δ)∗ + (φ ∗ + δ) ∗ = δ, where δ(x) = 1 2 x2. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.
AB - We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body K° or the dual function φ∗ play the role of the inverses "K-1" and "φ-1", we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function φ one has (φ + δ)∗ + (φ ∗ + δ) ∗ = δ, where δ(x) = 1 2 x2. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.
KW - Legendre transform
KW - Polarity
KW - geometric mean
KW - summand
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U2 - 10.1142/S0219199716500279
DO - 10.1142/S0219199716500279
M3 - Article
AN - SCOPUS:84976584088
SN - 0219-1997
VL - 18
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
IS - 6
M1 - 1650027
ER -