The average-distance problem, in the penalized formulation, involves minimizing
(1) Eμλ(Σ) := ∫ℝd d(x,Σ)dμ(x) + λℋ1(Σ),
among path-wise connected, closed sets Σ with finite ℋ1-measure, where
d ≥ 2, μ is a given measure, λ is a given parameter and
d(x,Σ) := infy∈Σ|x - y|.
The average-distance problem can be also considered among compact, convex sets with perimeter and/or
volume penalization, i.e. minimizing
(2) ℰ(μ,λ1,λ2)(·):=∫ℝd d(x,·)dμ(x) + λ1Per(·)
where μ is a given measure,
λ1,λ2 ≥ 0 are given parameters with
λ1 + λ2 > 0,
and the unknown varies among compact, convex sets. Very little
is known about the regularity of minimizers of (2). In particular it is unclear
if minimizers of (2) are in general C1 regular.
The aim of this paper is twofold: first, we provide in ℝ2
a second approach in constructing minimizers
of (1) which are not C1 regular; then, using the same technique,
we provide an example of minimizer of (2)
whose border is not C1 regular, under perimeter penalization only.