Cutting by half-spaces and the perimeter of convex sets #
Let μ be a measure on a finite-dimensional real inner product space E, finite on compact sets
(an additive Haar measure for the results on convex sets). This file proves that cutting a set of
finite measure by a closed half-space does not increase its perimeter, and deduces that
intersecting such a set with a convex set does not increase its perimeter either. In particular
the perimeter is monotone under inclusion of convex sets: a convex set has perimeter at most that
of any null-measurable set of finite measure containing it, and bounded convex sets have finite
perimeter. Cutting by the half-spaces bounding large cubes also shows that a set of finite
measure is the limit of bounded subsets whose perimeters converge to its own.
Monotonicity under convex inclusion is how perimeters of convex bodies are compared. For a
convex body K containing the ball closedBall c r, the outer parallel body
K + t • closedBall 0 1 lies in the dilate of K about c by 1 + t / r, so its perimeter is at
most (1 + t / r) ^ (n - 1) P(K). This is the upper bound needed to identify the right derivative
of the volume of outer parallel bodies with the perimeter.
The half-space estimate is proved by duality, without any boundary regularity. Given an admissible
test vector field φ, the field ψ that extends the normal component of φ on the hyperplane
{f = c} constantly in the normal direction is divergence-free above the hyperplane, so pairing
s with φ below the hyperplane and with ψ above it gives ∫_{s ∩ {f ≤ c}} div φ up to a
term controlled by μ (s ∩ {f ≥ c + n}). The glued field is not smooth, but it can be smoothed
across a slab of width 1 / (n + 1) because the normal components agree on the hyperplane
(TauCeti.ofReal_setIntegral_add_setIntegral_le_perimeter).
Main results #
TauCeti.ofReal_setIntegral_add_setIntegral_le_perimeter: two admissible test vector fields whose normal components agree on a hyperplane can be glued along it.TauCeti.perimeter_inter_halfSpace_le:P(s ∩ {f ≤ c}) ≤ P(s)forsof finite measure.TauCeti.perimeter_inter_biInter_halfSpace_le: the same for finitely many half-spaces.MeasureTheory.NullMeasurableSet.exists_isBounded_perimeter_approximation: bounded truncations of a set of finite measure, with perimeters at most and converging to its perimeter.Convex.perimeter_inter_le:P(s ∩ C) ≤ P(s)forCconvex andμ s < ∞.Convex.perimeter_le_of_subset:P(K) ≤ P(L)forKconvex andK ⊆ L,μ L < ∞.Convex.perimeter_lt_top: a bounded convex set has finite perimeter.
References #
- F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Chapters 12 and 13.
Gluing admissible vector fields along a hyperplane. Let φ and ψ be test vector fields
on Ω with ‖φ‖, ‖ψ‖ ≤ 1 whose normal components f ∘ φ and f ∘ ψ agree on the hyperplane
{f = a}. Then the field equal to φ on the half-space {f ≤ a} and to ψ on {f > a} still
pairs with a null-measurable set s to at most its perimeter:
∫_{s ∩ {f ≤ a}} div φ + ∫_{s ∩ {f > a}} div ψ ≤ P(s; Ω).
Cutting by a half-space does not increase the perimeter. For a null-measurable set s of
finite measure and a closed half-space {f ≤ c}, P(s ∩ {f ≤ c}) ≤ P(s).
The finiteness of μ s cannot be dropped: for an additive Haar measure the whole space has
perimeter zero, while a half-space {f ≤ c} with f ≠ 0 has positive perimeter (infinite in
dimension at least two).
Cutting by finitely many half-spaces does not increase the perimeter. For a
null-measurable set s of finite measure and finitely many closed half-spaces {g k ≤ c k},
k ∈ S, the perimeter of s ∩ ⋂ k ∈ S, {g k ≤ c k} is at most P(s).
Bounded truncation in perimeter. A null-measurable set s of finite measure is the limit
of an increasing sequence of bounded null-measurable subsets t n ⊆ s, converging to s in
measure, whose perimeters are at most P(s) and converge to P(s). The perimeters are
extended-valued, so s may have infinite perimeter. The finiteness of μ s cannot be dropped,
since bounded subsets of a set of infinite measure do not converge to it in measure.
Intersecting with a convex set does not increase the perimeter. For a convex set C and
a null-measurable set s of finite measure, P(s ∩ C) ≤ P(s).
The finiteness of μ s cannot be dropped: the complement of a small ball has small perimeter, but
its intersection with a large concentric ball has larger perimeter.
The perimeter is monotone under inclusion of a convex set. If a convex set K is
contained in a null-measurable set L of finite measure, then P(K) ≤ P(L). In particular the
perimeter is monotone on bounded convex sets.
The finiteness of μ L cannot be dropped: a half-space {f ≤ c} with f ≠ 0 is convex and has
positive perimeter (infinite in dimension at least two), while the whole space has perimeter
zero.
A bounded convex set has finite perimeter.