Riesz-type Inequalities and Overdetermined Problems for Triangles and Quadrilaterals

Abstract

Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in the literature involving other shape functionals as well as further open problems.

The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities

“…To that aim, it is convenient to reformulate more explicitly the shape derivative in the left hand side of (11). This has been done in [6], but to make the paper self-contained we enclose a proof in the Appendix, see Lemma 19. The outcome is the following: if {Ω ε } are obtained from Ω respectively by rotating the side [A i A i+1 ] with respect to its midpoint M i , or by a parallel movement of such side with respect to itself, the stationarity condition (11) amounts to ask that, setting v Ω (x) := Ω h(x − y)dy, it holds…”

“…See Theorem 2. In particular, the above mentioned conjecture made in [6] is, in general, false. The heuristic reason is that, for r large enough, our problem is equivalent to the problem of finding so-called "largest small Ngons", namely polygons with fixed diameter and maximal area.…”

“…To sketch the relation between the polygonal Riesz inequality and the physical isoperimetric inequalities (4), let us focus for simplicity on the Saint-Venant inequality. It is known that the torsional rigidity satisfies (see [17,Section 4]) (6) τ (Ω) = A probabilistic reformulation in the same vein can be given also for the principal frequency, see again [17,Section 4]. Then the classical Saint-Venant and Faber-Krahn inequalities, with the ball as optimal domain, can be obtained as a consequence of the general rearrangement inequality for multiple integrals due to Brascamp-Lieb-Luttinger [8, Theorem 1.2], which in 2-dimensions reads here h i are measurable non-negative functions on R 2 vanishing at infinity, h * i are their symmetric decreasing rearrangements, {a ij } are real numbers, Ω is a set of finite Lebesgue measure in R 2 , and Ω * is the ball with the same area as Ω.…”

“…Concerning the inequality (10), for N = 3, 4 (triangle and quadrilaterals), though not explicitly stated it can be deduced via Steiner symmetrization from Lemma 3.2 in [8]. More recently, in [6] Bonacini, Cristoferi and Topaloglu have considered the strictly related problem of characterizing "critical" triangles and quadrilaterals. More precisely, a polygon Ω ∈ P N is said to be critical for J h if, for some positive constant c, it holds (11)…”

“…Clearly, a polygon maximizing J h over P N must be a critical polygon. In [6] it is proved that, under some weak assumptions on h, the regular triangle and the square are respectively the unique critical triangle and quadrilateral. It is also conjectured that the same rigidity property holds true for any N ≥ 5 and that, consequently, the inequality (10) remains true for every N ≥ 5.…”