The L-Brunn-Minkowski inequality for p < 1

Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries

“…We note that understanding the equality case in Conjecture 1.2 clarifies the uniqueness of the solution of the Monge-Ampere type logarithmic Minkowski Problem (see Boroczky, Lutwak, Yang, Zhang [15], Kolesnikov, Milman [52], Chen, Huang, Li, Liu [20]).…”

“…After initiating the study of the L p -Brunn-Minkowski inequality for a range of p by Firey [36] and Lutwak [55,56,57], major results have been obtained by Hug, Lutwak, Yang, Zhang [46], and more recently the papers Kolesnikov, Milman [52], Chen, Huang, Li, Liu [20], Hosle, Kolesnikov, Livshyts [45], Kolesnikov, Livshyts [51] present new developments and approaches.…”

On the planar L-Minkowski problem

“…The non-unique result for p ∈ (−∞, −7) and constant function f ≡ 1 was given in part (2) of our Theorem 1.5. The final part (6) was known already owing to the explicitly examples as mention above.…”

“…More recently, Jian-Lu-Wang [20] have proven that for p ∈ (−n − 1, 0), there exists at least a smooth positive function f such that (1.1) admits two different solutions. While a partial uniqueness result was established by Chen-Huang-Li-Liu [6] on origin symmetric convex bodies for p ∈ (p 0 , 1), p 0 ∈ (0, 1), using the L p -Brunn-Minkowski inequality. For the deep negative case p ≤ −n − 1, the situations are more complicated.…”

The $L_p$-Gaussian Minkowski problem

“…Lutwak [37] testified there exists a unique o-symmetry convex body such that S p (K) = µ for p > 1. In case of 0 < p < 1, L p -Brunn-Minkowski inequality and its condition of the equality holds have not been improved much until [8]. While the logarithmic Minkowski problem (p = 0) and the centro-affine Minkowski problem (p = −n) are two special cases; see, e.g., [5], and [12].…”