DOI: 10.68381/jca1015 ISSN: 0944-6532
Convex Bodies of Optimal Shape
G. Carlier, T. Lachand-Robert
Given a continuous function
f\colon S^{n-1}\to\mathbb{R}
f
:
S
n
−
1
→
R
, we consider the minimization of the functional
\int_{\partial A} f(\nu_A)\,d\mathcal{H}^{n-1}
∫
∂
A
f
(
ν
A
)
d
H
n
−
1
with respect to the subset A of
\mathbb{R}^n
R
n
, included in a class of convex bodies defined by surface or shape conditions. This corresponds to non-parametric formulations of older problems, including Newton's problem of the body of minimal resistance, following an approach due to G. Buttazzo and P. Guasoni [J. Convex Analysis 4 (1997) 343–351]. We establish existence and uniqueness results and some characterizations of the minimizers