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1. Inequalities

    1. Log-Brunn-Minkowski Inequality

          For convex bodies K and L in \R^n with support functions h_K and h_L the logarithmic sum tK +_\circ (1-t)L is defined as tK +_\circ (1-t)L = \underset{u \in S^{n-1}}{\cap} \left\{x \in \R^n:\ \langle x, u\rangle \leq h_K(u)^th_L(u)^{1-t} \right\}
      where h_K(u) = \max\limits_{x \in K} \langle x, u\rangle .

      Add remark

      Conjecture 1.05.

      Show that if K and L symmetric convex bodies in \R^n, then for all 0 \leq t \leq 1 the following hods |tK +_\circ (1-t)L| \geq |K|^t |L|^{1-t}
      with equality if and only if K = L or K and L are parallelograms with parallel sides.
          The possible way to approach the problem is to find some functional versions for logarithmic Brunn-Minkowski inequality.

        • L_p-Brunn Minkowski inequality

              If K and L are convex bodies that contain the origin in their interiors then the L_p-combination tK +_p (1-t)L is defined by tK +_p (1-t)L = \underset{u \in S^{n-1}}{\cap} \left\{ x \in \R^n:\ \left< x , u \right> \leq \left(t h_K(u)^p + (1-t) h_L(u)^p \right)^{\frac1p} \right\}.

          Add remark

          Conjecture 1.1.

          Show that if K and L symmetric convex bodies in \R^n, then for all 0 \leq t \leq 1 and p\geq 0 |tK +_p (1-t)L| \geq |K|^t |L|^{1-t}.

            • Log-Minkowski inequality


              Add remark

              Problem 1.15.

              Show that if K and L symmetric convex bodies in \R^n, then \int\limits_{S^{n-1}} \log{\frac{h_K(u)}{h_L(u)}}\,d\bar{V}_L(u) \geq \frac1n \log{\frac{|K|}{|L|}},
              where \bar{V}_L is the cone-volume probability measure of L.

                • Cone-volume measure


                  Add remark

                  Problem 1.2.

                  If K and L symmetric convex bodies in \R^n such that \bar{V}_K=\bar{V}_L, then K=L or K and L are parallelograms with parallel sides.

                    • Brunn-Minkowski inequality for dual quermassintegrals


                      Add remark

                      Problem 1.25.

                      Show that for convex bodies K and L \widetilde{W}_{n-i}(K + L)^{\frac1i} \geq \widetilde{W}_{n-i}(K)^{\frac1i} + \widetilde{W}_{n-i}(L)^{\frac1i},
                      where \tilde{W}_{n-i}(K) = \int\limits_{S^{n-1}} \rho_K(u)^i\,du.

                        • Brunn-Minkowski inequality for L_p surface area


                          Add remark

                          Problem 1.3.

                          Show that for convex bodies K and L in \R^n and for 0 \leq p \leq 1 S_p(K+L)^{\frac1{n-p}} \geq S_p(K)^{\frac1{n-p}} + S_p(L)^{\frac1{n-p}},
                          where S_p(K) = \int\limits_{S^{n-1}} h_K(u)^{1-p} \, dS_K(u) is the L_p surface area.

                            • Conjecture due to D. Cordero-Erausquin


                              Add remark

                              Conjecture 1.35.

                              If K and L symmetric convex bodies in \R^n, then |K\cap L||K^\circ \cap L| \leq |B^n_2\cap L||B^n_2\cap L|,
                              where K^\circ =\{y \in \R^n:\ \forall x \in K \, \langle x ,y \rangle \leq 1 \} is a polar body of K.

                                • Dar’s conjecture


                                  Add remark

                                  Conjecture 1.4.

                                  Let \mu(K, L) is defined by \mu(K, L) = \max\limits_{x \in \R^n} |K \cap (x + L)|. Then for convex bodies K and L in \R^n |K + L|^{\frac1n} \geq \mu(K, L) + \frac{|K|^{\frac1n} |L|^{\frac1n}}{\mu(K, L)}.

                                    • Convex bodies in \mathbb{C}^n


                                      Add remark

                                      Problem 1.45.

                                      Consider a convex body in \mathbb{C}^n and fix a real subspace E of \mathbb{C}^n. Define V_E(K) = \int\limits_{U(n)} |K|\phi(E)| \, d\phi, where U(n) is a unitary group. Does Brunn-Minkowski type inequality hold for such quantities: V_E(K + L)^{\frac1i} \geq V_E(K)^{\frac1i} +V_E(L)^{\frac1i}?
                                      Also, for what type of subspaces E this will work?


                                        • Add remark

                                          Problem 1.5.

                                          Isoperimetric type problems for quermassintegrals in hyperbolic or spherical space.

                                              Cite this as: AimPL: Symmetry and convexity in geometric inequalities, available at http://aimpl.org/symconvgeomineq.