Jensen-type geometric shapes

We present both necessary and sufficient conditions to the convex closed shape $X$ such that the inequality $$ \frac{1}{|X|} \int_X f(x)\:dx \le \frac{1}{|\partial X|} \int_{\partial X} f(x)\:dx$$ is valid for every convex function $f \colon X \to \mathbb{R}$ ($\partial X$ stands for the boundary of $X$). It is proved that this inequality holds if $X$ is (i) an $n$-dimensional parallelotope, (ii) an $n$-dimensional ball, (iii) a convex polytope having an inscribed sphere (tangent to all its facets) with center in the center of mass of $\partial X$.


Introduction
Dragomir and Pearce proved [1, Theorem 215] that if B n is an ndimensional ball then for every convex function f ; here and below ffl stands for the average integral (more precisely ffl X f (x) dx := 1 |X|´X f (x) dx). During Conference on Inequalities and Applications 2016 Páles stated the problem whether for every convex and closed set X and every convex function f : X → R, the inequality is valid. It is however easy to verify that for the triangle T with vertices (0, −1), (0, 1), (1, 0) and the function f : T ∋ (x, y) → x, the inequality (1.1) voids (as the inequality 1 3 ≤ 1 − This motivates us to introduce the following definition. Convex and closed shape X is called Jensen-type if for every convex function f : X → R, the inequality (1.1) is satisfied.
Using this definition Dragomir-Pearce result can be expressed briefly as 3-dimensional ball is of Jensen-type or B 3 is of Jensen-type. The second example can be expressed by 45-45-90 triangle is not of Jensentype.
Motivated by these preliminaries we are going to prove this property for regular polygons, parallelotopes (in all dimensions), balls (in all dimensions), and Platonic solids.

Results
We begin with some necessary condition for X to be Jensen-type. It repeats the argumentation which was already presented in a case of triangle T .
Lemma 1. If X is of Jensen-type then centers of mass of X and ∂X coincide.
Proof. Let π i : R n → R be a projection on i-th coordinate (i ∈ {1, . . . , n}). Both π i and −π i are convex so, as X is of Jensen-type, we get . . , n}. But centers of mass of X and ∂X equal to , respectively. The above equality states that these points coincide.
Remark. We have presented some necessary condition for a shape to be of Jensen-type. Our conjecture is that every convex shape which satisfies this condition is of Jensen-type.
In the subsequent result we are going to prove that all parallelotopes and n-dimensional balls are of Jensen-type.
Proposition 1. All parallelotopes are of Jensen-type.
be its all facets. Denote by S * i the facet opposite to S i . In fact facet S * i is a facet S i shifted by some vector v i ∈ R n . Finally, for y ∈ S i , let y * := y+v i ∈ S * i . Now fix a convex function f : W → R. By Hermite-Hadamard inequality we havê We integrate both side over S i to obtain Finally, let us sum up the above inequality for i ∈ {1, 2, . . . , 2 n }. Then we obtain In the next proposition we will generalize the Dragomir-Pearce result.
Proposition 2. The n-dimensional ball is of Jensen-type for every n ≥ 2.
Proof. Fix a convex function f : B n → R. We have By the identity n|B n | = |S n−1 |, we obtain desired inequality.
2.1. Convex polytopes having an inscribed sphere. We will now struggle with convex polytopes. To avoid misunderstandings the inscribed sphere is the sphere which is tangent to all facets.
Lemma 2. Let n ∈ N, ∆ ⊂ R n be a convex (n − 1)-dimensional shape, s ∈ R n \ ∆ and G = conv{∆, s}. Then for every convex function Proof. For each θ ∈ (0, 1] let T θ a homothetic transformation of ∆ with center s and scale θ. Denote its image by ∆ θ . Moreover denote H := dist(p, ∆) and π : ∆ → ∆ 1 be a projection such that π| ∆ θ = T −1 θ . We know that Thus, by Jensen's and Fubini's inequalities, To finish the proof we can use the classical equality |G| = 1 n ·H ·|∆ 1 |. Theorem 1. Let W be an n-dimensional convex polytope having an inscribed sphere with center s. Then for every convex function f : W → R.
Corollary 1. Let W be a convex n-dimensional polytope having an inscribed sphere with center s and m be the center of mass of ∂W . Then for every convex function f : W → R.

Indeed, by the Jensen's inequality we have
We can now sum this inequality with (2.1) side-by-side to obtain desired inequality.
As a trivial particular case we obtain some sufficient condition for W to be of Jensen-type.
Theorem 2. Let W be a convex polytope having an inscribed sphere. If the center of this sphere coincide with the center of mass ∂W , then W is of Jensen-type.
Obviously this result implies that all Platonic solids are of Jensentype.