
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 $h_K(u) = \max\limits_{x \in K} \langle x, u\rangle .$

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 $$tK +_p (1-t)L = \underset{u \in S^{n-1}}{\cap} \left\{ x \in \R^n:\ \left< x , u \right> \leq \left(t h_K(u)^p + (1-t) h_L(u)^p \right)^{\frac1p} \right\}.$$

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

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

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

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

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

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

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 $\mathbb{C}^n$

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?
• 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.